id stringlengths 11 15 | title stringlengths 7 171 | problem stringlengths 9 4.33k | solution stringlengths 6 19k | problem_wikitext stringlengths 9 4.42k | solution_wikitext stringlengths 7 19.1k | proof_title stringlengths 9 171 | theorem_url stringlengths 34 198 | proof_url stringlengths 36 198 | categories listlengths 0 9 | theorem_references listlengths 0 36 | proof_references listlengths 0 253 |
|---|---|---|---|---|---|---|---|---|---|---|---|
proofwiki-1700 | Group of Order p q is Cyclic | Let $p, q$ be primes such that $p < q$ and $p$ does not divide $q - 1$.
Let $G$ be a group of order $p q$.
Then $G$ is cyclic. | Let the Sylow $p$-subgroup of $G$ be denoted $P$.
Let the Sylow $q$-subgroup of $G$ be denoted $Q$.
We have that:
:$P \cap Q = \set e$
where $e$ is the identity element of $G$.
Hence in $P \cup Q$ there are $q + p - 1$ elements.
As $p q \ge 2 q > q + p - 1$, there exists a non- identity element in $G$ that is not in $H... | Let $p, q$ be [[Definition:Prime Number|primes]] such that $p < q$ and $p$ does not [[Definition:Divisor of Integer|divide]] $q - 1$.
Let $G$ be a [[Definition:Group|group]] of [[Definition:Order of Structure|order]] $p q$.
Then $G$ is [[Definition:Cyclic Group|cyclic]]. | Let the [[Definition:Sylow p-Subgroup|Sylow $p$-subgroup]] of $G$ be denoted $P$.
Let the [[Definition:Sylow p-Subgroup|Sylow $q$-subgroup]] of $G$ be denoted $Q$.
We have that:
:$P \cap Q = \set e$
where $e$ is the [[Definition:Identity Element|identity element]] of $G$.
Hence in $P \cup Q$ there are $q + p - 1$ [... | Group of Order p q is Cyclic/Proof 2 | https://proofwiki.org/wiki/Group_of_Order_p_q_is_Cyclic | https://proofwiki.org/wiki/Group_of_Order_p_q_is_Cyclic/Proof_2 | [
"Group of Order p q is Cyclic",
"Finite Cyclic Groups",
"Groups of Order p q"
] | [
"Definition:Prime Number",
"Definition:Divisor (Algebra)/Integer",
"Definition:Group",
"Definition:Order of Structure",
"Definition:Cyclic Group"
] | [
"Definition:Sylow p-Subgroup",
"Definition:Sylow p-Subgroup",
"Definition:Identity (Abstract Algebra)/Two-Sided Identity",
"Definition:Element",
"Definition:Identity (Abstract Algebra)/Two-Sided Identity",
"Definition:Order of Group Element",
"Definition:Cyclic Group"
] |
proofwiki-1701 | Derivative of Logarithm at One | Let $\ln x$ be the natural logarithm of $x$ for real $x$ where $x > 0$.
Then:
:$\ds \lim_{x \mathop \to 0} \frac {\map \ln {1 + x} } x = 1$ | L'Hôpital's Rule gives:
:$\ds \lim_{x \mathop \to c} \frac {\map f x} {\map g x} = \lim_{x \mathop \to c} \frac {\map {f'} x} {\map {g'} x}$
(provided the appropriate conditions are fulfilled).
Here we have:
{{begin-eqn}}
{{eqn | l = \map \ln {1 + 0}
| r = 0
}}
{{eqn | l = \map {D_x} {\map \ln {1 + x} }
| r... | Let $\ln x$ be the [[Definition:Natural Logarithm|natural logarithm]] of $x$ for [[Definition:Real Number|real]] $x$ where $x > 0$.
Then:
:$\ds \lim_{x \mathop \to 0} \frac {\map \ln {1 + x} } x = 1$ | [[L'Hôpital's Rule]] gives:
:$\ds \lim_{x \mathop \to c} \frac {\map f x} {\map g x} = \lim_{x \mathop \to c} \frac {\map {f'} x} {\map {g'} x}$
(provided the appropriate conditions are fulfilled).
Here we have:
{{begin-eqn}}
{{eqn | l = \map \ln {1 + 0}
| r = 0
}}
{{eqn | l = \map {D_x} {\map \ln {1 + x} }
... | Derivative of Logarithm at One/Proof 1 | https://proofwiki.org/wiki/Derivative_of_Logarithm_at_One | https://proofwiki.org/wiki/Derivative_of_Logarithm_at_One/Proof_1 | [
"Derivative of Logarithm at One",
"Derivatives involving Logarithm Functions",
"Logarithms"
] | [
"Definition:Natural Logarithm",
"Definition:Real Number"
] | [
"L'Hôpital's Rule",
"Derivative of Composite Function",
"Derivative of Identity Function"
] |
proofwiki-1702 | Derivative of Logarithm at One | Let $\ln x$ be the natural logarithm of $x$ for real $x$ where $x > 0$.
Then:
:$\ds \lim_{x \mathop \to 0} \frac {\map \ln {1 + x} } x = 1$ | {{begin-eqn}}
{{eqn | l = \lim_{x \mathop \to 0} \frac {\map \ln {1 + x} } x
| r = \lim_{x \mathop \to 0} \frac {\map \ln {1 + x} - \ln 1} x
| c = subtract $\ln 1 = 0$ from the numerator, from Logarithm of 1 is 0
}}
{{eqn | r = \intlimits {\dfrac {\d} {\d x} \ln x} {x \mathop = 1} {}
| c = {{Defof|Der... | Let $\ln x$ be the [[Definition:Natural Logarithm|natural logarithm]] of $x$ for [[Definition:Real Number|real]] $x$ where $x > 0$.
Then:
:$\ds \lim_{x \mathop \to 0} \frac {\map \ln {1 + x} } x = 1$ | {{begin-eqn}}
{{eqn | l = \lim_{x \mathop \to 0} \frac {\map \ln {1 + x} } x
| r = \lim_{x \mathop \to 0} \frac {\map \ln {1 + x} - \ln 1} x
| c = subtract $\ln 1 = 0$ from the numerator, from [[Logarithm of 1 is 0]]
}}
{{eqn | r = \intlimits {\dfrac {\d} {\d x} \ln x} {x \mathop = 1} {}
| c = {{Defof... | Derivative of Logarithm at One/Proof 2 | https://proofwiki.org/wiki/Derivative_of_Logarithm_at_One | https://proofwiki.org/wiki/Derivative_of_Logarithm_at_One/Proof_2 | [
"Derivative of Logarithm at One",
"Derivatives involving Logarithm Functions",
"Logarithms"
] | [
"Definition:Natural Logarithm",
"Definition:Real Number"
] | [
"Natural Logarithm of 1 is 0",
"Derivative of Natural Logarithm Function"
] |
proofwiki-1703 | Derivative of Logarithm at One | Let $\ln x$ be the natural logarithm of $x$ for real $x$ where $x > 0$.
Then:
:$\ds \lim_{x \mathop \to 0} \frac {\map \ln {1 + x} } x = 1$ | Note that this proof does not presuppose Derivative of Natural Logarithm Function.
{{begin-eqn}}
{{eqn | l = \lim_{x \mathop \to 0} \frac {\map \ln {1 + x} } x
| r = \lim_{n \mathop \to \infty} \frac {\map \ln {1 + \frac 1 n} } {\frac 1 n}
| c =
}}
{{eqn | r = \lim_{n \mathop \to \infty} n \, \map \ln {1 +... | Let $\ln x$ be the [[Definition:Natural Logarithm|natural logarithm]] of $x$ for [[Definition:Real Number|real]] $x$ where $x > 0$.
Then:
:$\ds \lim_{x \mathop \to 0} \frac {\map \ln {1 + x} } x = 1$ | Note that this proof does not presuppose [[Derivative of Natural Logarithm Function]].
{{begin-eqn}}
{{eqn | l = \lim_{x \mathop \to 0} \frac {\map \ln {1 + x} } x
| r = \lim_{n \mathop \to \infty} \frac {\map \ln {1 + \frac 1 n} } {\frac 1 n}
| c =
}}
{{eqn | r = \lim_{n \mathop \to \infty} n \, \map \ln ... | Derivative of Logarithm at One/Proof 3 | https://proofwiki.org/wiki/Derivative_of_Logarithm_at_One | https://proofwiki.org/wiki/Derivative_of_Logarithm_at_One/Proof_3 | [
"Derivative of Logarithm at One",
"Derivatives involving Logarithm Functions",
"Logarithms"
] | [
"Definition:Natural Logarithm",
"Definition:Real Number"
] | [
"Derivative of Natural Logarithm Function",
"Natural Logarithm of e is 1"
] |
proofwiki-1704 | Dicyclic Group is Non-Abelian Group | The dicyclic group $\Dic n$ is a non-abelian group on two generators. | The dicyclic group $\Dic n$ is defined as follows:
{{:Definition:Dicyclic Group}}
First it is to be demonstrated that $\Dic n$ is a group.
First we deduce the following:
$(1): \quad b^4 = e$:
{{begin-eqn}}
{{eqn | l = b^2
| r = a^n
| c = {{Defof|Dicyclic Group}}
}}
{{eqn | ll= \leadsto
| l = \paren {b... | The [[Definition:Dicyclic Group|dicyclic group]] $\Dic n$ is a non-[[Definition:Abelian Group|abelian]] [[Definition:Group|group]] on two [[Definition:Generator of Group|generators]]. | The [[Definition:Dicyclic Group|dicyclic group]] $\Dic n$ is defined as follows:
{{:Definition:Dicyclic Group}}
First it is to be demonstrated that $\Dic n$ is a [[Definition:Group|group]].
First we deduce the following:
$(1): \quad b^4 = e$:
{{begin-eqn}}
{{eqn | l = b^2
| r = a^n
| c = {{Defof|Dicycli... | Dicyclic Group is Non-Abelian Group | https://proofwiki.org/wiki/Dicyclic_Group_is_Non-Abelian_Group | https://proofwiki.org/wiki/Dicyclic_Group_is_Non-Abelian_Group | [
"Dicyclic Groups"
] | [
"Definition:Dicyclic Group",
"Definition:Abelian Group",
"Definition:Group",
"Definition:Generator of Group"
] | [
"Definition:Dicyclic Group",
"Definition:Group",
"Definition:Group Product",
"Axiom:Group Axioms",
"Axiom:Group Axioms",
"Definition:Group"
] |
proofwiki-1705 | Existence of Interval of Convergence of Power Series | Let $\xi \in \R$ be a real number.
Let $\ds \map S x = \sum_{n \mathop = 0}^\infty a_n \paren {x - \xi}^n$ be a power series about $\xi$.
Then the interval of convergence of $\map S x$ is a real interval whose midpoint is $\xi$. | Suppose $\map S x$ converges when $x = y$.
We need to show that it converges for all $x$ which satisfy $\size {x - \xi} < \size {y - \xi}$.
So, let $\map S x$ converge when $x = y$.
Then from Terms in Convergent Series Converge to Zero:
:$a_n \paren {y - \xi}^n \to 0$ as $n \to \infty$
Hence, from Convergent Sequence i... | Let $\xi \in \R$ be a [[Definition:Real Number|real number]].
Let $\ds \map S x = \sum_{n \mathop = 0}^\infty a_n \paren {x - \xi}^n$ be a [[Definition:Power Series|power series]] about $\xi$.
Then the [[Definition:Interval of Convergence|interval of convergence]] of $\map S x$ is a [[Definition:Real Interval|real i... | Suppose $\map S x$ [[Definition:Convergent Series|converges]] when $x = y$.
We need to show that it converges for all $x$ which satisfy $\size {x - \xi} < \size {y - \xi}$.
So, let $\map S x$ converge when $x = y$.
Then from [[Terms in Convergent Series Converge to Zero]]:
:$a_n \paren {y - \xi}^n \to 0$ as $n \to ... | Existence of Interval of Convergence of Power Series | https://proofwiki.org/wiki/Existence_of_Interval_of_Convergence_of_Power_Series | https://proofwiki.org/wiki/Existence_of_Interval_of_Convergence_of_Power_Series | [
"Power Series",
"Radius of Convergence",
"Existence of Interval of Convergence of Power Series"
] | [
"Definition:Real Number",
"Definition:Power Series",
"Definition:Interval of Convergence",
"Definition:Real Interval",
"Definition:Real Interval/Midpoint"
] | [
"Definition:Convergent Series",
"Terms in Convergent Series Converge to Zero",
"Convergent Sequence in Metric Space is Bounded",
"Definition:Bounded Sequence",
"Sequence of Powers of Number less than One",
"Definition:Convergent Series",
"Comparison Test"
] |
proofwiki-1706 | Power Series is Differentiable on Interval of Convergence | Let $\xi \in \R$ be a real number.
Let $\ds \map f x = \sum_{n \mathop = 0}^\infty a_n \paren {x - \xi}^n$ be a (real) power series about $\xi$.
Let $\map f x$ have an interval of convergence $I$.
Then $\map f x$ is continuous on $I$, and differentiable on $I$ except possibly at its endpoints.
Also:
:$\ds \map {D_x} {\... | Let the radius of convergence of $\map f x$ be $R$.
Suppose $x \in I$ such that $x$ is not an endpoint of $I$.
Then there exists $x_0 \in I$ such that $x$ lies between $x_0$ and $\xi$.
Thus:
:$\size {x - \xi} < \size {x_0 - \xi} < R$
Consider the series:
:$\ds \sum_{n \mathop = 2}^\infty \size {\frac {n \paren {n - 1} ... | Let $\xi \in \R$ be a [[Definition:Real Number|real number]].
Let $\ds \map f x = \sum_{n \mathop = 0}^\infty a_n \paren {x - \xi}^n$ be a [[Definition:Real Power Series|(real) power series]] about $\xi$.
Let $\map f x$ have an [[Definition:Interval of Convergence|interval of convergence]] $I$.
Then $\map f x$ is [... | Let the [[Definition:Radius of Convergence|radius of convergence]] of $\map f x$ be $R$.
Suppose $x \in I$ such that $x$ is not an [[Definition:Endpoint of Real Interval|endpoint]] of $I$.
Then there exists $x_0 \in I$ such that $x$ lies between $x_0$ and $\xi$.
Thus:
:$\size {x - \xi} < \size {x_0 - \xi} < R$
Con... | Power Series is Differentiable on Interval of Convergence | https://proofwiki.org/wiki/Power_Series_is_Differentiable_on_Interval_of_Convergence | https://proofwiki.org/wiki/Power_Series_is_Differentiable_on_Interval_of_Convergence | [
"Real Power Series",
"Convergence",
"Differential Calculus"
] | [
"Definition:Real Number",
"Definition:Power Series/Real Domain",
"Definition:Interval of Convergence",
"Definition:Continuous Real Function/Interval",
"Definition:Differentiable Mapping/Real Function/Interval",
"Definition:Real Interval/Endpoints"
] | [
"Definition:Radius of Convergence",
"Definition:Real Interval/Endpoints",
"Radius of Convergence from Limit of Sequence",
"Difference of Two Powers",
"Closed Form for Triangular Numbers",
"Definition:Differentiable Mapping/Real Function/Point",
"Definition:Real Interval/Endpoints",
"Definition:Continu... |
proofwiki-1707 | Power Series Expansion for Exponential Function | Let $\exp x$ be the exponential function.
Then:
{{begin-eqn}}
{{eqn | q = \forall x \in \R
| l = \exp x
| r = \sum_{n \mathop = 0}^\infty \frac {x^n} {n!}
| c =
}}
{{eqn | r = 1 + x + \frac {x^2} {2!} + \frac {x^3} {3!} + \cdots
| c =
}}
{{end-eqn}} | From Higher Derivatives of Exponential Function, we have:
:$\forall n \in \N: \map {f^{\paren n} } {\exp x} = \exp x$
Since $\exp 0 = 1$, the Taylor series expansion for $\exp x$ about $0$ is given by:
:$\ds \exp x = \sum_{n \mathop = 0}^\infty \frac {x^n} {n!}$
From Radius of Convergence of Power Series over Factorial... | Let $\exp x$ be the [[Definition:Real Exponential Function|exponential function]].
Then:
{{begin-eqn}}
{{eqn | q = \forall x \in \R
| l = \exp x
| r = \sum_{n \mathop = 0}^\infty \frac {x^n} {n!}
| c =
}}
{{eqn | r = 1 + x + \frac {x^2} {2!} + \frac {x^3} {3!} + \cdots
| c =
}}
{{end-eqn}} | From [[Higher Derivatives of Exponential Function]], we have:
:$\forall n \in \N: \map {f^{\paren n} } {\exp x} = \exp x$
Since $\exp 0 = 1$, the [[Definition:Taylor Series|Taylor series]] expansion for $\exp x$ about $0$ is given by:
:$\ds \exp x = \sum_{n \mathop = 0}^\infty \frac {x^n} {n!}$
From [[Radius of Con... | Power Series Expansion for Exponential Function | https://proofwiki.org/wiki/Power_Series_Expansion_for_Exponential_Function | https://proofwiki.org/wiki/Power_Series_Expansion_for_Exponential_Function | [
"Power Series Expansion for Exponential Function",
"Exponential Series",
"Exponential Function",
"Examples of Power Series",
"Taylor Series"
] | [
"Definition:Exponential Function/Real"
] | [
"Higher Derivatives of Exponential Function",
"Definition:Taylor Series",
"Radius of Convergence of Power Series over Factorial",
"Definition:Power Series",
"Taylor's Theorem",
"Exponential is Strictly Increasing",
"Series of Power over Factorial Converges"
] |
proofwiki-1708 | Derivative of Cosine Function | :$\map {\dfrac \d {\d x} } {\cos x} = -\sin x$ | From the definition of the cosine function, we have:
:$\ds \cos x = \sum_{n \mathop = 0}^\infty \paren {-1}^n \frac {x^{2 n} } {\paren {2 n}!}$
Then:
{{begin-eqn}}
{{eqn | l = \map {\frac \d {\d x} } {\cos x}
| r = \sum_{n \mathop = 1}^\infty \paren {-1}^n 2 n \frac {x^{2 n - 1} } {\paren {2 n}!}
| c = Powe... | :$\map {\dfrac \d {\d x} } {\cos x} = -\sin x$ | From the definition of the [[Definition:Real Cosine Function|cosine function]], we have:
:$\ds \cos x = \sum_{n \mathop = 0}^\infty \paren {-1}^n \frac {x^{2 n} } {\paren {2 n}!}$
Then:
{{begin-eqn}}
{{eqn | l = \map {\frac \d {\d x} } {\cos x}
| r = \sum_{n \mathop = 1}^\infty \paren {-1}^n 2 n \frac {x^{2 n -... | Derivative of Cosine Function/Proof 1 | https://proofwiki.org/wiki/Derivative_of_Cosine_Function | https://proofwiki.org/wiki/Derivative_of_Cosine_Function/Proof_1 | [
"Derivatives of Trigonometric Functions",
"Cosine Function",
"Derivative of Cosine Function"
] | [] | [
"Definition:Cosine/Real Function",
"Power Series is Differentiable on Interval of Convergence",
"Definition:Sine/Real Function"
] |
proofwiki-1709 | Derivative of Cosine Function | :$\map {\dfrac \d {\d x} } {\cos x} = -\sin x$ | {{begin-eqn}}
{{eqn | l = \map {\frac \d {\d x} } {\cos x}
| r = \lim_{h \mathop \to 0} \frac {\map \cos {x + h} - \cos x} h
| c = {{Defof|Derivative of Real Function at Point}}
}}
{{eqn | r = \lim_{h \mathop \to 0} \frac {\cos x \cos h - \sin x \sin h - \cos x} h
| c = Cosine of Sum
}}
{{eqn | r = \l... | :$\map {\dfrac \d {\d x} } {\cos x} = -\sin x$ | {{begin-eqn}}
{{eqn | l = \map {\frac \d {\d x} } {\cos x}
| r = \lim_{h \mathop \to 0} \frac {\map \cos {x + h} - \cos x} h
| c = {{Defof|Derivative of Real Function at Point}}
}}
{{eqn | r = \lim_{h \mathop \to 0} \frac {\cos x \cos h - \sin x \sin h - \cos x} h
| c = [[Cosine of Sum]]
}}
{{eqn | r ... | Derivative of Cosine Function/Proof 2 | https://proofwiki.org/wiki/Derivative_of_Cosine_Function | https://proofwiki.org/wiki/Derivative_of_Cosine_Function/Proof_2 | [
"Derivatives of Trigonometric Functions",
"Cosine Function",
"Derivative of Cosine Function"
] | [] | [
"Cosine of Sum",
"Combination Theorem for Limits of Functions/Real/Sum Rule",
"Combination Theorem for Limits of Functions/Real/Multiple Rule",
"Limit of (Cosine (X) - 1) over X at Zero",
"Limit of Sinc Function at Zero"
] |
proofwiki-1710 | Derivative of Cosine Function | :$\map {\dfrac \d {\d x} } {\cos x} = -\sin x$ | {{begin-eqn}}
{{eqn | l = \frac \d {\d x} \cos x
| r = \frac \d {\d x} \map \sin {\frac \pi 2 - x}
| c = Sine of Complement equals Cosine
}}
{{eqn | r = -\map \cos {\frac \pi 2 - x}
| c = Derivative of Sine Function and Chain Rule for Derivatives
}}
{{eqn | r = -\sin x
| c = Cosine of Complemen... | :$\map {\dfrac \d {\d x} } {\cos x} = -\sin x$ | {{begin-eqn}}
{{eqn | l = \frac \d {\d x} \cos x
| r = \frac \d {\d x} \map \sin {\frac \pi 2 - x}
| c = [[Sine of Complement equals Cosine]]
}}
{{eqn | r = -\map \cos {\frac \pi 2 - x}
| c = [[Derivative of Sine Function]] and [[Chain Rule for Derivatives]]
}}
{{eqn | r = -\sin x
| c = [[Cosin... | Derivative of Cosine Function/Proof 3 | https://proofwiki.org/wiki/Derivative_of_Cosine_Function | https://proofwiki.org/wiki/Derivative_of_Cosine_Function/Proof_3 | [
"Derivatives of Trigonometric Functions",
"Cosine Function",
"Derivative of Cosine Function"
] | [] | [
"Sine of Complement equals Cosine",
"Derivative of Sine Function",
"Derivative of Composite Function",
"Cosine of Complement equals Sine"
] |
proofwiki-1711 | Derivative of Cosine Function | :$\map {\dfrac \d {\d x} } {\cos x} = -\sin x$ | {{begin-eqn}}
{{eqn | l = \map {\frac \d {\d x} } {\cos x}
| r = \lim_{h \mathop \to 0} \frac {\map \cos {x + h} - \cos x} h
| c = {{Defof|Derivative of Real Function at Point}}
}}
{{eqn | r = \lim_{h \mathop \to 0} \frac {\map \cos {\paren {x + \frac h 2} + \frac h 2} - \map \cos {\paren {x + \frac h 2} - ... | :$\map {\dfrac \d {\d x} } {\cos x} = -\sin x$ | {{begin-eqn}}
{{eqn | l = \map {\frac \d {\d x} } {\cos x}
| r = \lim_{h \mathop \to 0} \frac {\map \cos {x + h} - \cos x} h
| c = {{Defof|Derivative of Real Function at Point}}
}}
{{eqn | r = \lim_{h \mathop \to 0} \frac {\map \cos {\paren {x + \frac h 2} + \frac h 2} - \map \cos {\paren {x + \frac h 2} - ... | Derivative of Cosine Function/Proof 4 | https://proofwiki.org/wiki/Derivative_of_Cosine_Function | https://proofwiki.org/wiki/Derivative_of_Cosine_Function/Proof_4 | [
"Derivatives of Trigonometric Functions",
"Cosine Function",
"Derivative of Cosine Function"
] | [] | [
"Werner Formulas/Sine by Sine",
"Combination Theorem for Limits of Functions/Real/Multiple Rule",
"Combination Theorem for Limits of Functions/Real/Product Rule",
"Real Sine Function is Continuous",
"Limit of Sinc Function at Zero"
] |
proofwiki-1712 | Derivative of Cosine Function | :$\map {\dfrac \d {\d x} } {\cos x} = -\sin x$ | {{begin-eqn}}
{{eqn | l = 1
| r = \cos^2 x + \sin^2 x
| c = Sum of Squares of Sine and Cosine
}}
{{eqn | ll= \leadsto
| l = \map {D_x} 1
| r = \map {D_x} {\cos^2 x + \sin^2 x}
| c = differentiating both sides
}}
{{eqn | ll= \leadsto
| l = 0
| r = 2 \map {D_x} {\cos x} \cos x +... | :$\map {\dfrac \d {\d x} } {\cos x} = -\sin x$ | {{begin-eqn}}
{{eqn | l = 1
| r = \cos^2 x + \sin^2 x
| c = [[Sum of Squares of Sine and Cosine]]
}}
{{eqn | ll= \leadsto
| l = \map {D_x} 1
| r = \map {D_x} {\cos^2 x + \sin^2 x}
| c = differentiating both sides
}}
{{eqn | ll= \leadsto
| l = 0
| r = 2 \map {D_x} {\cos x} \cos ... | Derivative of Cosine Function/Proof 5 | https://proofwiki.org/wiki/Derivative_of_Cosine_Function | https://proofwiki.org/wiki/Derivative_of_Cosine_Function/Proof_5 | [
"Derivatives of Trigonometric Functions",
"Cosine Function",
"Derivative of Cosine Function"
] | [] | [
"Sum of Squares of Sine and Cosine",
"Product Rule for Derivatives",
"Derivative of Composite Function",
"Derivative of Sine Function"
] |
proofwiki-1713 | Limit of Sinc Function at Zero | :$\ds \lim_{x \mathop \to 0} \frac {\sin x} x = 1$ | {{begin-eqn}}
{{eqn | l = \sin x
| r = \sum_{n \mathop = 0}^\infty \paren {-1}^n \frac {x^{2 n + 1} } {\paren {2 n + 1}!}
| c = {{Defof|Real Sine Function}}
}}
{{eqn | r = \paren {-1}^0 \frac {x^{2 \cdot 0 + 1} } {\paren {2 \cdot 0 + 1}!} + \sum_{n \mathop = 1}^\infty \paren {-1}^n \frac {x^{2 n + 1} } {\pa... | :$\ds \lim_{x \mathop \to 0} \frac {\sin x} x = 1$ | {{begin-eqn}}
{{eqn | l = \sin x
| r = \sum_{n \mathop = 0}^\infty \paren {-1}^n \frac {x^{2 n + 1} } {\paren {2 n + 1}!}
| c = {{Defof|Real Sine Function}}
}}
{{eqn | r = \paren {-1}^0 \frac {x^{2 \cdot 0 + 1} } {\paren {2 \cdot 0 + 1}!} + \sum_{n \mathop = 1}^\infty \paren {-1}^n \frac {x^{2 n + 1} } {\pa... | Limit of Sinc Function at Zero/Proof 1 | https://proofwiki.org/wiki/Limit_of_Sinc_Function_at_Zero | https://proofwiki.org/wiki/Limit_of_Sinc_Function_at_Zero/Proof_1 | [
"Limit of Sinc Function at Zero",
"Sinc Function",
"Sine Function",
"Differential Calculus",
"Examples of Limits of Real Functions"
] | [] | [
"Power Series is Differentiable on Interval of Convergence",
"L'Hôpital's Rule",
"Real Polynomial Function is Continuous"
] |
proofwiki-1714 | Limit of Sinc Function at Zero | :$\ds \lim_{x \mathop \to 0} \frac {\sin x} x = 1$ | From Sine of Zero is Zero:
:$\sin 0 = 0$
From Derivative of Sine Function:
:$\map {D_x} {\sin x} = \cos x$
Then by Cosine of Zero is One:
:$\cos 0 = 1$
From Derivative of Identity Function:
:$\map {D_x} x = 1$
Thus L'Hôpital's Rule applies and so:
:$\ds \lim_{x \mathop \to 0} \frac {\sin x} x = \lim_{x \mathop \to 0} \... | :$\ds \lim_{x \mathop \to 0} \frac {\sin x} x = 1$ | From [[Sine of Zero is Zero]]:
:$\sin 0 = 0$
From [[Derivative of Sine Function]]:
:$\map {D_x} {\sin x} = \cos x$
Then by [[Cosine of Zero is One]]:
:$\cos 0 = 1$
From [[Derivative of Identity Function]]:
:$\map {D_x} x = 1$
Thus [[L'Hôpital's Rule]] applies and so:
:$\ds \lim_{x \mathop \to 0} \frac {\sin x} x =... | Limit of Sinc Function at Zero/Proof 2 | https://proofwiki.org/wiki/Limit_of_Sinc_Function_at_Zero | https://proofwiki.org/wiki/Limit_of_Sinc_Function_at_Zero/Proof_2 | [
"Limit of Sinc Function at Zero",
"Sinc Function",
"Sine Function",
"Differential Calculus",
"Examples of Limits of Real Functions"
] | [] | [
"Sine of Zero is Zero",
"Derivative of Sine Function",
"Cosine of Zero is One",
"Derivative of Identity Function",
"L'Hôpital's Rule"
] |
proofwiki-1715 | Sum of Squares of Sine and Cosine | :$\cos^2 x + \sin^2 x = 1$ | {{begin-eqn}}
{{eqn | l = 1
| r = \cos 0
| c = Cosine of Zero is One
}}
{{eqn | r = \map \cos {x - x}
| c =
}}
{{eqn | r = \cos x \map \cos {-x} - \sin x \map \sin {-x}
| c = Cosine of Sum
}}
{{eqn | r = \cos x \cos x - \paren {-\sin x \sin x}
| c = Cosine Function is Even and Sine Functi... | :$\cos^2 x + \sin^2 x = 1$ | {{begin-eqn}}
{{eqn | l = 1
| r = \cos 0
| c = [[Cosine of Zero is One]]
}}
{{eqn | r = \map \cos {x - x}
| c =
}}
{{eqn | r = \cos x \map \cos {-x} - \sin x \map \sin {-x}
| c = [[Cosine of Sum]]
}}
{{eqn | r = \cos x \cos x - \paren {-\sin x \sin x}
| c = [[Cosine Function is Even]] and... | Sum of Squares of Sine and Cosine/Proof 1 | https://proofwiki.org/wiki/Sum_of_Squares_of_Sine_and_Cosine | https://proofwiki.org/wiki/Sum_of_Squares_of_Sine_and_Cosine/Proof_1 | [
"Sum of Squares of Sine and Cosine",
"Sine Function",
"Cosine Function"
] | [] | [
"Cosine of Zero is One",
"Cosine of Sum",
"Cosine Function is Even",
"Sine Function is Odd"
] |
proofwiki-1716 | Sum of Squares of Sine and Cosine | :$\cos^2 x + \sin^2 x = 1$ | From the trigonometric definitions of sine and cosine:
{{begin-eqn}}
{{eqn | l = \sin x
| r = \frac{\text{opposite} } {\text{hypotenuse} }
}}
{{eqn | l = \cos x
| r = \frac{\text{adjacent} } {\text{hypotenuse} }
}}
{{end-eqn}}
Then:
{{begin-eqn}}
{{eqn | l = \sin^2 x + \cos^2 x
| r = \frac{\text{oppos... | :$\cos^2 x + \sin^2 x = 1$ | From the trigonometric definitions of [[Definition:Sine of Angle|sine]] and [[Definition:Cosine of Angle|cosine]]:
{{begin-eqn}}
{{eqn | l = \sin x
| r = \frac{\text{opposite} } {\text{hypotenuse} }
}}
{{eqn | l = \cos x
| r = \frac{\text{adjacent} } {\text{hypotenuse} }
}}
{{end-eqn}}
Then:
{{begin-eqn}}... | Sum of Squares of Sine and Cosine/Proof 2 | https://proofwiki.org/wiki/Sum_of_Squares_of_Sine_and_Cosine | https://proofwiki.org/wiki/Sum_of_Squares_of_Sine_and_Cosine/Proof_2 | [
"Sum of Squares of Sine and Cosine",
"Sine Function",
"Cosine Function"
] | [] | [
"Definition:Sine/Definition from Triangle",
"Definition:Cosine/Definition from Triangle",
"Definition:Square/Function",
"Pythagoras's Theorem"
] |
proofwiki-1717 | Sum of Squares of Sine and Cosine | :$\cos^2 x + \sin^2 x = 1$ | Let $P = \tuple {x, y}$ be a point on the circumference of a unit circle whose center is at the origin of a cartesian plane.
From Sine of Angle in Cartesian Plane and Cosine of Angle in Cartesian Plane:
:$P = \tuple {\cos \theta, \sin \theta}$
The graph of the unit circle is the locus of:
:$x^2 + y^2 = 1$
as given by E... | :$\cos^2 x + \sin^2 x = 1$ | Let $P = \tuple {x, y}$ be a [[Definition:Point|point]] on the [[Definition:Circumference of Circle|circumference]] of a [[Definition:Unit Circle|unit circle]] whose [[Definition:Center of Circle|center]] is at the [[Definition:Origin|origin]] of a [[Definition:Cartesian Plane|cartesian plane]].
From [[Sine of Angle ... | Sum of Squares of Sine and Cosine/Proof 3 | https://proofwiki.org/wiki/Sum_of_Squares_of_Sine_and_Cosine | https://proofwiki.org/wiki/Sum_of_Squares_of_Sine_and_Cosine/Proof_3 | [
"Sum of Squares of Sine and Cosine",
"Sine Function",
"Cosine Function"
] | [] | [
"Definition:Point",
"Definition:Circle/Circumference",
"Definition:Unit Circle",
"Definition:Circle/Center",
"Definition:Coordinate System/Origin",
"Definition:Cartesian Plane",
"Sine of Angle in Cartesian Plane",
"Cosine of Angle in Cartesian Plane",
"Definition:Relation/Graph",
"Definition:Unit ... |
proofwiki-1718 | Sum of Squares of Sine and Cosine | :$\cos^2 x + \sin^2 x = 1$ | {{begin-eqn}}
{{eqn | l = \cos^2 x + \sin^2 x
| r = \paren {\cos x + i \, \sin x} \paren {\cos x - i \, \sin x}
| c = factoring over the complex numbers
}}
{{eqn | r = \paren {\cos x + i \, \sin x} \paren {\map \cos {-x} + i \, \map \sin {-x} }
| c = Cosine Function is Even and Sine Function is Odd
}}... | :$\cos^2 x + \sin^2 x = 1$ | {{begin-eqn}}
{{eqn | l = \cos^2 x + \sin^2 x
| r = \paren {\cos x + i \, \sin x} \paren {\cos x - i \, \sin x}
| c = factoring over the [[Definition:Complex Number|complex numbers]]
}}
{{eqn | r = \paren {\cos x + i \, \sin x} \paren {\map \cos {-x} + i \, \map \sin {-x} }
| c = [[Cosine Function is ... | Sum of Squares of Sine and Cosine/Proof 4 | https://proofwiki.org/wiki/Sum_of_Squares_of_Sine_and_Cosine | https://proofwiki.org/wiki/Sum_of_Squares_of_Sine_and_Cosine/Proof_4 | [
"Sum of Squares of Sine and Cosine",
"Sine Function",
"Cosine Function"
] | [] | [
"Definition:Complex Number",
"Cosine Function is Even",
"Sine Function is Odd",
"Euler's Formula"
] |
proofwiki-1719 | Sum of Squares of Sine and Cosine | :$\cos^2 x + \sin^2 x = 1$ | {{begin-eqn}}
{{eqn | l = \cos^2 x + \sin^2 x
| r = \paren {\frac {e^{i x} + e^{-i x} } 2}^2 + \sin^2 x
| c = Euler's Cosine Identity
}}
{{eqn | r = \paren {\frac {e^{i x} + e^{-i x} } 2}^2 + \paren {\frac {e^{i x} - e^{-i x} } {2 i} }^2
| c = Euler's Sine Identity
}}
{{eqn | r = \frac {\paren {e^{i x... | :$\cos^2 x + \sin^2 x = 1$ | {{begin-eqn}}
{{eqn | l = \cos^2 x + \sin^2 x
| r = \paren {\frac {e^{i x} + e^{-i x} } 2}^2 + \sin^2 x
| c = [[Euler's Cosine Identity]]
}}
{{eqn | r = \paren {\frac {e^{i x} + e^{-i x} } 2}^2 + \paren {\frac {e^{i x} - e^{-i x} } {2 i} }^2
| c = [[Euler's Sine Identity]]
}}
{{eqn | r = \frac {\paren... | Sum of Squares of Sine and Cosine/Proof 5 | https://proofwiki.org/wiki/Sum_of_Squares_of_Sine_and_Cosine | https://proofwiki.org/wiki/Sum_of_Squares_of_Sine_and_Cosine/Proof_5 | [
"Sum of Squares of Sine and Cosine",
"Sine Function",
"Cosine Function"
] | [] | [
"Euler's Cosine Identity",
"Euler's Sine Identity",
"Square of Sum",
"Square of Difference",
"Exponential of Sum"
] |
proofwiki-1720 | Cosine of Sum | :$\map \cos {a + b} = \cos a \cos b - \sin a \sin b$ | {{begin-eqn}}
{{eqn | l = \map \cos {a + b} + i \, \map \sin {a + b}
| r = e^{i \paren {a + b} }
| c = Euler's Formula
}}
{{eqn | r = e^{i a} e^{i b}
| c = Exponential of Sum
}}
{{eqn | r = \paren {\cos a + i \sin a} \paren {\cos b + i \sin b}
| c = Euler's Formula
}}
{{eqn | r = \paren {\cos a ... | :$\map \cos {a + b} = \cos a \cos b - \sin a \sin b$ | {{begin-eqn}}
{{eqn | l = \map \cos {a + b} + i \, \map \sin {a + b}
| r = e^{i \paren {a + b} }
| c = [[Euler's Formula]]
}}
{{eqn | r = e^{i a} e^{i b}
| c = [[Exponential of Sum]]
}}
{{eqn | r = \paren {\cos a + i \sin a} \paren {\cos b + i \sin b}
| c = [[Euler's Formula]]
}}
{{eqn | r = \pa... | Cosine of Sum/Proof 1 | https://proofwiki.org/wiki/Cosine_of_Sum | https://proofwiki.org/wiki/Cosine_of_Sum/Proof_1 | [
"Cosine of Sum",
"Cosine Function",
"Trigonometric Addition Formulas"
] | [] | [
"Euler's Formula",
"Exponential of Sum",
"Euler's Formula",
"Complex Numbers form Field",
"Definition:Complex Number/Real Part"
] |
proofwiki-1721 | Cosine of Sum | :$\map \cos {a + b} = \cos a \cos b - \sin a \sin b$ | Recall the analytic definitions of sine and cosine:
:$\ds \sin x = \sum_{n \mathop = 0}^\infty \paren {-1}^n \frac {x^{2 n + 1} } {\paren {2 n + 1}!}$
:$\ds \cos x = \sum_{n \mathop = 0}^\infty \paren {-1}^n \frac {x^{2 n} } {\paren {2 n}!}$
Let:
{{begin-eqn}}
{{eqn | l = \map g a
| r = \map \sin {a + b} - \sin a... | :$\map \cos {a + b} = \cos a \cos b - \sin a \sin b$ | Recall the analytic definitions of [[Definition:Real Sine Function|sine]] and [[Definition:Real Cosine Function|cosine]]:
:$\ds \sin x = \sum_{n \mathop = 0}^\infty \paren {-1}^n \frac {x^{2 n + 1} } {\paren {2 n + 1}!}$
:$\ds \cos x = \sum_{n \mathop = 0}^\infty \paren {-1}^n \frac {x^{2 n} } {\paren {2 n}!}$
Let:
... | Cosine of Sum/Proof 2 | https://proofwiki.org/wiki/Cosine_of_Sum | https://proofwiki.org/wiki/Cosine_of_Sum/Proof_2 | [
"Cosine of Sum",
"Cosine Function",
"Trigonometric Addition Formulas"
] | [] | [
"Definition:Sine/Real Function",
"Definition:Cosine/Real Function",
"Definition:Differentiation",
"Derivative of Sine Function",
"Derivative of Cosine Function",
"Derivative of Constant",
"Square of Real Number is Non-Negative"
] |
proofwiki-1722 | Cosine of Sum | :$\map \cos {a + b} = \cos a \cos b - \sin a \sin b$ | {{begin-eqn}}
{{eqn | l = \cos a \cos b - \sin a \sin b
| r = \paren {\frac {e^{i a} + e^{-i a} } 2} \paren {\frac {e^{i b} + e^{-i b} } 2} - \sin a \sin b
| c = Euler's Cosine Identity
}}
{{eqn | r = \paren {\frac {e^{i a} + e^{-i a} } 2} \paren {\frac {e^{i b} + e^{-i b} } 2} - \paren {\frac {e^{i a} - e... | :$\map \cos {a + b} = \cos a \cos b - \sin a \sin b$ | {{begin-eqn}}
{{eqn | l = \cos a \cos b - \sin a \sin b
| r = \paren {\frac {e^{i a} + e^{-i a} } 2} \paren {\frac {e^{i b} + e^{-i b} } 2} - \sin a \sin b
| c = [[Euler's Cosine Identity]]
}}
{{eqn | r = \paren {\frac {e^{i a} + e^{-i a} } 2} \paren {\frac {e^{i b} + e^{-i b} } 2} - \paren {\frac {e^{i a}... | Cosine of Sum/Proof 3 | https://proofwiki.org/wiki/Cosine_of_Sum | https://proofwiki.org/wiki/Cosine_of_Sum/Proof_3 | [
"Cosine of Sum",
"Cosine Function",
"Trigonometric Addition Formulas"
] | [] | [
"Euler's Cosine Identity",
"Euler's Sine Identity",
"Exponential of Sum",
"Euler's Cosine Identity"
] |
proofwiki-1723 | Cosine of Sum | :$\map \cos {a + b} = \cos a \cos b - \sin a \sin b$ | :File:Tri1.PNG
$AB$, $AC$, $AE$, and $AD$ are radii of the circle centered at $A$.
Let $\angle BAC = a$ and $\angle DAC = \angle BAE = b$.
By Euclid's First Postulate, we can construct line segments $BD$ and $CE$.
By Euclid's second common notion, $\angle DAB = \angle CAE$.
Thus by Triangle Side-Angle-Side Congruence, ... | :$\map \cos {a + b} = \cos a \cos b - \sin a \sin b$ | :[[File:Tri1.PNG]]
$AB$, $AC$, $AE$, and $AD$ are [[Definition:Radius of Circle|radii]] of the [[Definition:Circle|circle]] centered at $A$.
Let $\angle BAC = a$ and $\angle DAC = \angle BAE = b$.
By [[Axiom:Euclid's First Postulate|Euclid's First Postulate]], we can construct [[Definition:Line Segment|line segment... | Cosine of Sum/Proof 4 | https://proofwiki.org/wiki/Cosine_of_Sum | https://proofwiki.org/wiki/Cosine_of_Sum/Proof_4 | [
"Cosine of Sum",
"Cosine Function",
"Trigonometric Addition Formulas"
] | [] | [
"File:Tri1.PNG",
"Definition:Circle/Radius",
"Definition:Circle",
"Axiom:Euclid's First Postulate",
"Definition:Line/Segment",
"Axiom:Euclid's Common Notions",
"Triangle Side-Angle-Side Congruence",
"Definition:Cartesian Coordinate System",
"Cosine Function is Even",
"Sine Function is Odd",
"Def... |
proofwiki-1724 | Cosine of Sum | :$\map \cos {a + b} = \cos a \cos b - \sin a \sin b$ | :900px
We begin by enclosing a right-angled triangle $BEF$ with hypotenuse $BF$ of length $1$, inside Rectangle $ABCD$.
Let $\angle EBF = a$ and $\angle ABE = b$.
Therefore:
{{begin-eqn}}
{{eqn | l = BF
| r = 1
| c = Given
}}
{{eqn | l = BE
| r = \cos a
| c = {{Defof|Cosine of Angle}}
}}
{{eqn |... | :$\map \cos {a + b} = \cos a \cos b - \sin a \sin b$ | :[[File:Angle-sum.png|900px]]
We begin by enclosing a [[Definition:Right-Angled Triangle|right-angled triangle]] $BEF$ with [[Definition:Hypotenuse|hypotenuse]] $BF$ of length $1$, inside [[Definition:Rectangle|Rectangle]] $ABCD$.
Let $\angle EBF = a$ and $\angle ABE = b$.
Therefore:
{{begin-eqn}}
{{eqn | l = BF
... | Cosine of Sum/Proof 5 | https://proofwiki.org/wiki/Cosine_of_Sum | https://proofwiki.org/wiki/Cosine_of_Sum/Proof_5 | [
"Cosine of Sum",
"Cosine Function",
"Trigonometric Addition Formulas"
] | [] | [
"File:Angle-sum.png",
"Definition:Triangle (Geometry)/Right-Angled",
"Definition:Triangle (Geometry)/Right-Angled/Hypotenuse",
"Definition:Quadrilateral/Rectangle"
] |
proofwiki-1725 | Cosine of Sum | :$\map \cos {a + b} = \cos a \cos b - \sin a \sin b$ | {{begin-eqn}}
{{eqn | l = \map \cos {a + b}
| r = \map \cos {a - \paren {-b} }
| c =
}}
{{eqn | r = \cos a \map \cos {-b} + \sin a \map \sin {-b}
| c = Cosine of Difference
}}
{{eqn | r = \cos a \cos b + \sin a \paren {-\sin b}
| c = Cosine Function is Even, Sine Function is Odd
}}
{{eqn | r = ... | :$\map \cos {a + b} = \cos a \cos b - \sin a \sin b$ | {{begin-eqn}}
{{eqn | l = \map \cos {a + b}
| r = \map \cos {a - \paren {-b} }
| c =
}}
{{eqn | r = \cos a \map \cos {-b} + \sin a \map \sin {-b}
| c = [[Cosine of Difference]]
}}
{{eqn | r = \cos a \cos b + \sin a \paren {-\sin b}
| c = [[Cosine Function is Even]], [[Sine Function is Odd]]
}}
... | Cosine of Sum/Proof 6 | https://proofwiki.org/wiki/Cosine_of_Sum | https://proofwiki.org/wiki/Cosine_of_Sum/Proof_6 | [
"Cosine of Sum",
"Cosine Function",
"Trigonometric Addition Formulas"
] | [] | [
"Cosine of Difference",
"Cosine Function is Even",
"Sine Function is Odd"
] |
proofwiki-1726 | Cosine of Sum | :$\map \cos {a + b} = \cos a \cos b - \sin a \sin b$ | 400px
Let two triangles $\triangle ABC$ and $\triangle ABD$ be inscribed in a circle in the same semicircle:.
By Thales' Theorem, these are both right triangles with:
:$ \angle ACB = \angle ADB = 90 \degrees$
Let $AB = 1$.
Join $DC$.
By construction, $\Box ABCD$ is a cyclic quadrilateral.
Let:
:$\angle CAB = \alpha$
:$... | :$\map \cos {a + b} = \cos a \cos b - \sin a \sin b$ | [[File:Cosine of sum of angles.png|400px]]
Let two [[Definition:Triangle (Geometry)|triangles]] $\triangle ABC$ and $\triangle ABD$ be inscribed in a [[Definition:Circle|circle]] in the same [[Definition:Semicircle|semicircle]]:.
By [[Thales' Theorem]], these are both [[Definition:Right Triangle|right triangles]] wit... | Cosine of Sum/Proof 7 | https://proofwiki.org/wiki/Cosine_of_Sum | https://proofwiki.org/wiki/Cosine_of_Sum/Proof_7 | [
"Cosine of Sum",
"Cosine Function",
"Trigonometric Addition Formulas"
] | [] | [
"File:Cosine of sum of angles.png",
"Definition:Triangle (Geometry)",
"Definition:Circle",
"Definition:Circle/Semicircle",
"Thales' Theorem",
"Definition:Triangle (Geometry)/Right-Angled",
"Definition:Cyclic Quadrilateral",
"Length of Chord of Circle",
"Cosine of Complement equals Sine",
"Definiti... |
proofwiki-1727 | Characterisation of Sine and Cosine | Let $\map s x: \R \to \R$, $\map c x: \R \to \R$ be differentiable real functions that satisfy:
:$(1): \quad \map {s'} x = \map c x$
:$(2): \quad \map {c'} x = -\map s x$
:$(3): \quad \map s 0 = 0$
:$(4): \quad \map c 0 = 1$
:$(5): \quad \forall x: \map {s^2} x + \map {c^2} x = 1$
Then, for every $x \in \R$:
:$\map s x... | Define:
:$\map h x = \paren {\map c x - \map \cos x}^2 + \paren {\map s x - \map \sin x}^2$
Then:
{{begin-eqn}}
{{eqn | l = \map h x
| r = \map {c^2} x - 2 \map c x \map \cos x + \map {\cos^2} x + \map {s^2} x - 2 \map s x \map \sin x + \map {\sin^2} x
| c = Square of Difference
}}
{{eqn | r = 2 - 2 \paren ... | Let $\map s x: \R \to \R$, $\map c x: \R \to \R$ be [[Definition:Differentiable Real Function|differentiable real functions]] that satisfy:
:$(1): \quad \map {s'} x = \map c x$
:$(2): \quad \map {c'} x = -\map s x$
:$(3): \quad \map s 0 = 0$
:$(4): \quad \map c 0 = 1$
:$(5): \quad \forall x: \map {s^2} x + \map {c^2} ... | Define:
:$\map h x = \paren {\map c x - \map \cos x}^2 + \paren {\map s x - \map \sin x}^2$
Then:
{{begin-eqn}}
{{eqn | l = \map h x
| r = \map {c^2} x - 2 \map c x \map \cos x + \map {\cos^2} x + \map {s^2} x - 2 \map s x \map \sin x + \map {\sin^2} x
| c = [[Square of Difference]]
}}
{{eqn | r = 2 - 2 \p... | Characterisation of Sine and Cosine | https://proofwiki.org/wiki/Characterisation_of_Sine_and_Cosine | https://proofwiki.org/wiki/Characterisation_of_Sine_and_Cosine | [
"Sine Function",
"Cosine Function"
] | [
"Definition:Differentiable Mapping/Real Function"
] | [
"Square of Difference",
"Sum of Squares of Sine and Cosine",
"Linear Combination of Derivatives",
"Derivative of Constant",
"Product Rule for Derivatives",
"Derivative of Cosine Function",
"Derivative of Sine Function",
"Zero Derivative implies Constant Function",
"Cosine of Zero is One",
"Sine of... |
proofwiki-1728 | Real Cosine Function is Bounded | :$\size {\cos x} \le 1$ | From the algebraic definition of the real cosine function:
:$\ds \cos x = \sum_{n \mathop = 0}^\infty \paren {-1}^n \frac {x^{2 n} } {\paren {2 n}!}$
it follows that $\cos x$ is a real function.
Similarly, $\sin x$ is also a real function.
Thus it follows that:
{{begin-eqn}}
{{eqn | l = \cos^2 x
| o = \le
|... | :$\size {\cos x} \le 1$ | From the algebraic definition of the [[Definition:Real Cosine Function|real cosine function]]:
:$\ds \cos x = \sum_{n \mathop = 0}^\infty \paren {-1}^n \frac {x^{2 n} } {\paren {2 n}!}$
it follows that $\cos x$ is a [[Definition:Real Function|real function]].
Similarly, $\sin x$ is also a [[Definition:Real Function|... | Real Cosine Function is Bounded | https://proofwiki.org/wiki/Real_Cosine_Function_is_Bounded | https://proofwiki.org/wiki/Real_Cosine_Function_is_Bounded | [
"Cosine Function"
] | [] | [
"Definition:Cosine/Real Function",
"Definition:Real Function",
"Definition:Real Function",
"Square of Real Number is Non-Negative",
"Sum of Squares of Sine and Cosine",
"Ordering of Squares in Reals",
"Definition:Absolute Value"
] |
proofwiki-1729 | Boundedness of Sine X over X | Let $x \in \R$.
Then:
:$\size {\dfrac {\sin x} x} \le 1$ | From Derivative of Sine Function, we have:
:$D_x \paren {\sin x} = \cos x$
So by the Mean Value Theorem, there exists $\xi \in \R$ between $0$ and $x$ such that:
:$\dfrac {\sin x - \sin 0} {x - 0} = \cos \xi$
From Real Cosine Function is Bounded we have that:
:$\size {\cos \xi} \le 1$
{{qed}} | Let $x \in \R$.
Then:
:$\size {\dfrac {\sin x} x} \le 1$ | From [[Derivative of Sine Function]], we have:
:$D_x \paren {\sin x} = \cos x$
So by the [[Mean Value Theorem]], there exists $\xi \in \R$ between $0$ and $x$ such that:
:$\dfrac {\sin x - \sin 0} {x - 0} = \cos \xi$
From [[Real Cosine Function is Bounded]] we have that:
:$\size {\cos \xi} \le 1$
{{qed}} | Boundedness of Sine X over X | https://proofwiki.org/wiki/Boundedness_of_Sine_X_over_X | https://proofwiki.org/wiki/Boundedness_of_Sine_X_over_X | [
"Sine Function"
] | [] | [
"Derivative of Sine Function",
"Mean Value Theorem",
"Real Cosine Function is Bounded"
] |
proofwiki-1730 | Sine and Cosine are Periodic on Reals | The real sine function and real cosine function are periodic on the set of real numbers $\R$: | From the Real Cosine Function is Periodic and Real Sine Function is Periodic, we have that $\cos x$ and $\sin x$ are periodic on $\R$ with the same period.
If we denote the period of $\cos x$ and $\sin x$ as $p$, it follows that $\pi = \dfrac p 2$ is uniquely defined.
{{qed}} | The [[Definition:Real Sine Function|real sine function]] and [[Definition:Real Cosine Function|real cosine function]] are [[Definition:Periodic Real Function|periodic]] on the set of [[Definition:Real Number|real numbers]] $\R$: | From the [[Real Cosine Function is Periodic]] and [[Real Sine Function is Periodic]], we have that $\cos x$ and $\sin x$ are [[Definition:Periodic Real Function|periodic]] on $\R$ with the same [[Definition:Period of Periodic Real Function|period]].
If we denote the [[Definition:Period of Periodic Real Function|period... | Sine and Cosine are Periodic on Reals/Pi/Proof 1 | https://proofwiki.org/wiki/Sine_and_Cosine_are_Periodic_on_Reals | https://proofwiki.org/wiki/Sine_and_Cosine_are_Periodic_on_Reals/Pi/Proof_1 | [
"Sine and Cosine are Periodic on Reals",
"Sine Function",
"Cosine Function",
"Periodic Functions"
] | [
"Definition:Sine/Real Function",
"Definition:Cosine/Real Function",
"Definition:Periodic Function/Real",
"Definition:Real Number"
] | [
"Sine and Cosine are Periodic on Reals/Cosine",
"Sine and Cosine are Periodic on Reals/Sine",
"Definition:Periodic Function/Real",
"Definition:Periodic Real Function/Period",
"Definition:Periodic Real Function/Period"
] |
proofwiki-1731 | Sine and Cosine are Periodic on Reals | The real sine function and real cosine function are periodic on the set of real numbers $\R$: | By Cosine of Zero is One:
:$\cos 0 = 1$
By Cosine of 2 is Strictly Negative:
:$\cos 2 < 0$
Thus by {{Corollary|Intermediate Value Theorem}} there exists an $h \in \openint 0 2$ such that:
:$\cos h = 0$
By Sine of Sum for all $x \in \R$:
{{begin-eqn}}
{{eqn | l = \sin x
| r = \map \sin {x - h} \cos h + \map \cos {... | The [[Definition:Real Sine Function|real sine function]] and [[Definition:Real Cosine Function|real cosine function]] are [[Definition:Periodic Real Function|periodic]] on the set of [[Definition:Real Number|real numbers]] $\R$: | By [[Cosine of Zero is One]]:
:$\cos 0 = 1$
By [[Cosine of 2 is Strictly Negative]]:
:$\cos 2 < 0$
Thus by {{Corollary|Intermediate Value Theorem}} there exists an $h \in \openint 0 2$ such that:
:$\cos h = 0$
By [[Sine of Sum]] for all $x \in \R$:
{{begin-eqn}}
{{eqn | l = \sin x
| r = \map \sin {x - h} \co... | Sine and Cosine are Periodic on Reals/Pi/Proof 2 | https://proofwiki.org/wiki/Sine_and_Cosine_are_Periodic_on_Reals | https://proofwiki.org/wiki/Sine_and_Cosine_are_Periodic_on_Reals/Pi/Proof_2 | [
"Sine and Cosine are Periodic on Reals",
"Sine Function",
"Cosine Function",
"Periodic Functions"
] | [
"Definition:Sine/Real Function",
"Definition:Cosine/Real Function",
"Definition:Periodic Function/Real",
"Definition:Real Number"
] | [
"Cosine of Zero is One",
"Cosine of 2 is Strictly Negative",
"Sine of Sum",
"Cosine of Sum",
"Sum of Squares of Sine and Cosine",
"Definition:Periodic Function/Real",
"Nonconstant Periodic Function with no Period is Discontinuous Everywhere",
"Definition:Periodic Real Function/Period",
"Definition:P... |
proofwiki-1732 | Sine and Cosine are Periodic on Reals | The real sine function and real cosine function are periodic on the set of real numbers $\R$: | Since Real Cosine Function is Periodic, let $K$ be its period.
Then:
:$\cos K = \map \cos {0 + K} = \cos 0$
Because Cosine of Zero is One:
:$\cos K = 1$
Furthermore:
{{begin-eqn}}
{{eqn | l = \cos^2 K + \sin^2 K
| r = 1
| c = Sum of Squares of Sine and Cosine
}}
{{eqn | l = \sin^2 K
| r = 0
| c ... | The [[Definition:Real Sine Function|real sine function]] and [[Definition:Real Cosine Function|real cosine function]] are [[Definition:Periodic Real Function|periodic]] on the set of [[Definition:Real Number|real numbers]] $\R$: | Since [[Real Cosine Function is Periodic]], let $K$ be its [[Definition:Period of Periodic Real Function|period]].
Then:
:$\cos K = \map \cos {0 + K} = \cos 0$
Because [[Cosine of Zero is One]]:
:$\cos K = 1$
Furthermore:
{{begin-eqn}}
{{eqn | l = \cos^2 K + \sin^2 K
| r = 1
| c = [[Sum of Squares of Si... | Sine and Cosine are Periodic on Reals/Sine/Proof 1 | https://proofwiki.org/wiki/Sine_and_Cosine_are_Periodic_on_Reals | https://proofwiki.org/wiki/Sine_and_Cosine_are_Periodic_on_Reals/Sine/Proof_1 | [
"Sine and Cosine are Periodic on Reals",
"Sine Function",
"Cosine Function",
"Periodic Functions"
] | [
"Definition:Sine/Real Function",
"Definition:Cosine/Real Function",
"Definition:Periodic Function/Real",
"Definition:Real Number"
] | [
"Sine and Cosine are Periodic on Reals/Cosine",
"Definition:Periodic Real Function/Period",
"Cosine of Zero is One",
"Sum of Squares of Sine and Cosine",
"Sine of Sum",
"Definition:Periodic Function/Real",
"Definition:Periodic Real Function/Period",
"Definition:Periodic Real Function/Period",
"Sine ... |
proofwiki-1733 | Sine and Cosine are Periodic on Reals | The real sine function and real cosine function are periodic on the set of real numbers $\R$: | Since Real Cosine Function is Periodic, let $L$ be its period.
From Primitive of Cosine Function:
{{:Primitive of Cosine Function}}
for any constant $C$.
Therefore $\sin x$ is a Primitive of $\cos x$, for the special case of $C = 0$.
From Primitive of Periodic Real Function, it follows that $\sin x$ is periodic with pe... | The [[Definition:Real Sine Function|real sine function]] and [[Definition:Real Cosine Function|real cosine function]] are [[Definition:Periodic Real Function|periodic]] on the set of [[Definition:Real Number|real numbers]] $\R$: | Since [[Real Cosine Function is Periodic]], let $L$ be its [[Definition:Period of Periodic Real Function|period]].
From [[Primitive of Cosine Function]]:
{{:Primitive of Cosine Function}}
for any constant $C$.
Therefore $\sin x$ is a [[Definition:Primitive (Calculus)|Primitive]] of $\cos x$, for the special case of $... | Sine and Cosine are Periodic on Reals/Sine/Proof 2 | https://proofwiki.org/wiki/Sine_and_Cosine_are_Periodic_on_Reals | https://proofwiki.org/wiki/Sine_and_Cosine_are_Periodic_on_Reals/Sine/Proof_2 | [
"Sine and Cosine are Periodic on Reals",
"Sine Function",
"Cosine Function",
"Periodic Functions"
] | [
"Definition:Sine/Real Function",
"Definition:Cosine/Real Function",
"Definition:Periodic Function/Real",
"Definition:Real Number"
] | [
"Sine and Cosine are Periodic on Reals/Cosine",
"Definition:Periodic Real Function/Period",
"Primitive of Cosine Function",
"Definition:Primitive (Calculus)",
"Primitive of Periodic Real Function",
"Definition:Periodic Function/Real",
"Definition:Periodic Real Function/Period"
] |
proofwiki-1734 | Differentiable Bounded Convex Real Function is Constant | Let $f$ be a real function which is:
:$(1): \quad$ differentiable on $\R$
:$(2): \quad$ bounded on $\R$
:$(3): \quad$ convex on $\R$.
Then $f$ is constant. | Let $f$ be differentiable and bounded on $\R$.
Let $f$ be convex on $\R$.
Let $\xi \in \R$.
{{AimForCont}} $\map {f'} \xi > 0$.
Then by Mean Value of Convex Real Function it follows that:
:$\map f x \ge \map f \xi + \map {f'} \xi \paren {x - \xi} \to + \infty$ as $x \to +\infty$
and therefore is not bounded.
Similarly,... | Let $f$ be a [[Definition:Real Function|real function]] which is:
:$(1): \quad$ [[Definition:Everywhere Differentiable Real Function|differentiable]] on $\R$
:$(2): \quad$ [[Definition:Bounded Real-Valued Function|bounded]] on $\R$
:$(3): \quad$ [[Definition:Convex Real Function|convex]] on $\R$.
Then $f$ is [[Defini... | Let $f$ be [[Definition:Everywhere Differentiable Real Function|differentiable]] and [[Definition:Bounded Real-Valued Function|bounded]] on $\R$.
Let $f$ be [[Definition:Convex Real Function|convex]] on $\R$.
Let $\xi \in \R$.
{{AimForCont}} $\map {f'} \xi > 0$.
Then by [[Mean Value of Convex Real Function]] it fol... | Differentiable Bounded Convex Real Function is Constant | https://proofwiki.org/wiki/Differentiable_Bounded_Convex_Real_Function_is_Constant | https://proofwiki.org/wiki/Differentiable_Bounded_Convex_Real_Function_is_Constant | [
"Convex Real Functions",
"Differential Calculus"
] | [
"Definition:Real Function",
"Definition:Differentiable Mapping/Real Function/Real Number Line",
"Definition:Bounded Mapping/Real-Valued",
"Definition:Convex Real Function",
"Definition:Constant Mapping"
] | [
"Definition:Differentiable Mapping/Real Function/Real Number Line",
"Definition:Bounded Mapping/Real-Valued",
"Definition:Convex Real Function",
"Mean Value of Convex Real Function",
"Definition:Bounded Mapping/Real-Valued",
"Mean Value of Convex Real Function",
"Definition:Bounded Mapping/Real-Valued",... |
proofwiki-1735 | Banach-Tarski Paradox | Let $\mathbb D^3 \subset \R^3$ be a unit ball in real Euclidean space of $3$ dimensions.
Then $\mathbb D^3$ is equidecomposable to the union of two such unit balls. | Let $\mathbb D^3$ be centered at the origin, and $D^3$ be some other unit ball in $\R^3$ such that $\mathbb D^3 \cap D^3 = \O$.
Let $\mathbb S^2 = \partial \mathbb D^3$.
By the Hausdorff Paradox, there exists a decomposition of $ \mathbb S^2$ into four sets $A, B, C, Q$ such that $A, B, C$ and $B \cup C$ are congruent,... | Let $\mathbb D^3 \subset \R^3$ be a [[Definition:Unit Disk|unit]] [[Definition:Open Ball|ball]] in [[Definition:Real Euclidean Space|real Euclidean space]] of [[Definition:Dimension of Vector Space|$3$ dimensions]].
Then $\mathbb D^3$ is [[Definition:Equidecomposable Sets|equidecomposable]] to the [[Definition:Set Uni... | Let $\mathbb D^3$ be centered at the [[Definition:Origin|origin]], and $D^3$ be some other [[Definition:Unit Ball|unit ball]] in $\R^3$ such that $\mathbb D^3 \cap D^3 = \O$.
Let $\mathbb S^2 = \partial \mathbb D^3$.
By the [[Hausdorff Paradox]], there exists a [[Definition:Decomposition (Topology)|decomposition]] of... | Banach-Tarski Paradox/Proof 1 | https://proofwiki.org/wiki/Banach-Tarski_Paradox | https://proofwiki.org/wiki/Banach-Tarski_Paradox/Proof_1 | [
"Banach-Tarski Paradox",
"Equidecomposable Sets",
"Veridical Paradoxes",
"Topology"
] | [
"Definition:Unit Disk",
"Definition:Open Ball",
"Definition:Euclidean Space/Real",
"Definition:Dimension of Vector Space",
"Definition:Equidecomposable Sets",
"Definition:Set Union",
"Definition:Unit Disk",
"Definition:Open Ball"
] | [
"Definition:Coordinate System/Origin",
"Definition:Unit Ball",
"Hausdorff Paradox",
"Definition:Decomposable Set",
"Definition:Congruence (Metric Spaces)",
"Definition:Countable Set",
"Definition:Set",
"Definition:Equidecomposable Sets",
"Definition:Equidecomposable Sets",
"Definition:Equidecompos... |
proofwiki-1736 | Banach-Tarski Paradox | Let $\mathbb D^3 \subset \R^3$ be a unit ball in real Euclidean space of $3$ dimensions.
Then $\mathbb D^3$ is equidecomposable to the union of two such unit balls. | === Lemmata ===
{{:Banach-Tarski Paradox/Lemmata}}
Let $U$ denote the closed ball in real Euclidean space of $3$ dimensions defined as:
:$U = \set {\mathbf x \in \R^3 : \size {\mathbf x - \mathbf c} \le r}$
where:
:$r$ is the radius of $U$
:$\mathbf x$ is the position vector of a point $\tuple {x_1, x_2, x_3} \in \R^3$... | Let $\mathbb D^3 \subset \R^3$ be a [[Definition:Unit Disk|unit]] [[Definition:Open Ball|ball]] in [[Definition:Real Euclidean Space|real Euclidean space]] of [[Definition:Dimension of Vector Space|$3$ dimensions]].
Then $\mathbb D^3$ is [[Definition:Equidecomposable Sets|equidecomposable]] to the [[Definition:Set Uni... | === [[Banach-Tarski Paradox/Lemmata|Lemmata]] ===
{{:Banach-Tarski Paradox/Lemmata}}
Let $U$ denote the [[Definition:Closed Ball of Metric Space|closed ball]] in [[Definition:Real Euclidean Space|real Euclidean space]] of [[Definition:Dimension of Vector Space|$3$ dimensions]] defined as:
:$U = \set {\mathbf x \in \R... | Banach-Tarski Paradox/Proof 2 | https://proofwiki.org/wiki/Banach-Tarski_Paradox | https://proofwiki.org/wiki/Banach-Tarski_Paradox/Proof_2 | [
"Banach-Tarski Paradox",
"Equidecomposable Sets",
"Veridical Paradoxes",
"Topology"
] | [
"Definition:Unit Disk",
"Definition:Open Ball",
"Definition:Euclidean Space/Real",
"Definition:Dimension of Vector Space",
"Definition:Equidecomposable Sets",
"Definition:Set Union",
"Definition:Unit Disk",
"Definition:Open Ball"
] | [
"Banach-Tarski Paradox/Lemmata",
"Definition:Closed Ball/Metric Space",
"Definition:Euclidean Space/Real",
"Definition:Dimension of Vector Space",
"Definition:Closed Ball/Metric Space/Radius",
"Definition:Position Vector",
"Definition:Point",
"Definition:Position Vector",
"Definition:Closed Ball/Met... |
proofwiki-1737 | Equidecomposability is Equivalence Relation | The property of being equidecomposable is an equivalence relation on the power set $\powerset {\R^n}$. | === Relexivity ===
A set is necessarily equidecomposable with itself; the same decomposition and set of isometries suffice for $A$ as for $A$. | The property of being [[Definition:Equidecomposable Sets|equidecomposable]] is an [[Definition:Equivalence Relation|equivalence relation]] on the [[Definition:Power Set|power set]] $\powerset {\R^n}$. | === Relexivity ===
A set is necessarily [[Definition:Equidecomposable Sets|equidecomposable]] with itself; the same [[Definition:Decomposition (Topology)|decomposition]] and set of [[Definition:Isometry (Metric Spaces)|isometries]] suffice for $A$ as for $A$. | Equidecomposability is Equivalence Relation | https://proofwiki.org/wiki/Equidecomposability_is_Equivalence_Relation | https://proofwiki.org/wiki/Equidecomposability_is_Equivalence_Relation | [
"Equidecomposable Sets",
"Topology",
"Examples of Equivalence Relations"
] | [
"Definition:Equidecomposable Sets",
"Definition:Equivalence Relation",
"Definition:Power Set"
] | [
"Definition:Equidecomposable Sets",
"Definition:Decomposable Set",
"Definition:Isometry (Metric Spaces)",
"Definition:Equidecomposable Sets",
"Definition:Equidecomposable Sets",
"Definition:Equidecomposable Sets",
"Definition:Decomposable Set",
"Definition:Isometry (Metric Spaces)",
"Definition:Isom... |
proofwiki-1738 | Equidecomposability Unaffected by Union | {{explain|I guess the sets should be disjoint, in the sense that $S_i \cap S_j$ is empty for $i \ne j$. Similar for $T$.}}
Let $\set {S_1, \ldots, S_m}, \set {T_1, \ldots, T_m}$ be sets of sets in $\R^n$ such that:
:for each $k \in \set {1, \dots, m}, S_k$ and $T_k$ are equidecomposable.
Then the set $\ds S = \bigcup_... | We have for each $k \in \set{1, \dots, m}$ a decomposition $\set {A_{k, 1}, \cdots, A_{k, l_k} }$ and set of isometries $\phi_{i, j}: \R^n \to \R^n$ such that:
:$\ds S_k = \bigcup_{a \mathop = 1}^{l_k} \map {\phi_{k, a} } {A_{k, a} }$
and similarly for $T_k$ and some isometries $\theta_{i, j}: \R^n \to \R^n$:
:$\ds T_k... | {{explain|I guess the sets should be disjoint, in the sense that $S_i \cap S_j$ is empty for $i \ne j$. Similar for $T$.}}
Let $\set {S_1, \ldots, S_m}, \set {T_1, \ldots, T_m}$ be [[Definition:Set of Sets|sets of sets]] in $\R^n$ such that:
:for each $k \in \set {1, \dots, m}, S_k$ and $T_k$ are [[Definition:Equideco... | We have for each $k \in \set{1, \dots, m}$ a [[Definition:Decomposition (Topology)|decomposition]] $\set {A_{k, 1}, \cdots, A_{k, l_k} }$ and set of [[Definition:Isometry (Metric Spaces)|isometries]] $\phi_{i, j}: \R^n \to \R^n$ such that:
:$\ds S_k = \bigcup_{a \mathop = 1}^{l_k} \map {\phi_{k, a} } {A_{k, a} }$
and... | Equidecomposability Unaffected by Union | https://proofwiki.org/wiki/Equidecomposability_Unaffected_by_Union | https://proofwiki.org/wiki/Equidecomposability_Unaffected_by_Union | [
"Equidecomposable Sets",
"Topology"
] | [
"Definition:Set of Sets",
"Definition:Equidecomposable Sets",
"Definition:Equidecomposable Sets"
] | [
"Definition:Decomposable Set",
"Definition:Isometry (Metric Spaces)",
"Definition:Isometry (Metric Spaces)",
"Definition:Decomposable Set",
"Category:Equidecomposable Sets",
"Category:Topology"
] |
proofwiki-1739 | Subsets of Equidecomposable Subsets are Equidecomposable | Let $A, B \subseteq \R^n$ be equidecomposable.
Let $S \subseteq A$.
Then there exists $T \subseteq B$ such that $S$ and $T$ are equidecomposable. | Let $X_1, \dots, X_m$ be a decomposition of $A, B$ together with isometries $\mu_1, \ldots, \mu_m, \nu_1, \ldots, \nu_m: \R^n \to \R^n$ such that:
:$\ds A = \bigcup_{i \mathop = 1}^m \map {\mu_i} {X_i}$
and
:$\ds B = \bigcup_{i \mathop = 1}^m \map {\nu_i} {X_i}$
Define:
:$Y_i = \mu_i^{-1} \paren {S \cap \map {\mu_i} ... | Let $A, B \subseteq \R^n$ be [[Definition:Equidecomposable Sets|equidecomposable]].
Let $S \subseteq A$.
Then there exists $T \subseteq B$ such that $S$ and $T$ are [[Definition:Equidecomposable Sets|equidecomposable]]. | Let $X_1, \dots, X_m$ be a [[Definition:Decomposition (Topology)|decomposition]] of $A, B$ together with [[Definition:Isometry (Metric Spaces)|isometries]] $\mu_1, \ldots, \mu_m, \nu_1, \ldots, \nu_m: \R^n \to \R^n$ such that:
:$\ds A = \bigcup_{i \mathop = 1}^m \map {\mu_i} {X_i}$
and
:$\ds B = \bigcup_{i \mathop... | Subsets of Equidecomposable Subsets are Equidecomposable | https://proofwiki.org/wiki/Subsets_of_Equidecomposable_Subsets_are_Equidecomposable | https://proofwiki.org/wiki/Subsets_of_Equidecomposable_Subsets_are_Equidecomposable | [
"Equidecomposable Sets",
"Topology"
] | [
"Definition:Equidecomposable Sets",
"Definition:Equidecomposable Sets"
] | [
"Definition:Decomposable Set",
"Definition:Isometry (Metric Spaces)",
"Intersection Distributes over Union/Family of Sets",
"Intersection with Subset is Subset",
"Definition:Decomposable Set",
"Preimage of Subset is Subset of Preimage",
"Set Union Preserves Subsets",
"Category:Equidecomposable Sets",
... |
proofwiki-1740 | Shape of Cosine Function | The cosine function is:
:$(1): \quad$ strictly decreasing on the interval $\closedint 0 \pi$
:$(2): \quad$ strictly increasing on the interval $\closedint \pi {2 \pi}$
:$(3): \quad$ concave on the interval $\closedint {-\dfrac \pi 2} {\dfrac \pi 2}$
:$(4): \quad$ convex on the interval $\closedint {\dfrac \pi 2} {\dfra... | {{improve|Extract those results into separate pages and reference them. Referencing the "discussion" is unsound because the proof can be changed and that "discussion" removed.}}
From the discussion of Real Cosine Function is Periodic, we know that:
:$\cos x \ge 0$ on the closed interval $\closedint {-\dfrac \pi 2} {\df... | The [[Definition:Cosine|cosine]] function is:
:$(1): \quad$ [[Definition:Strictly Decreasing Real Function|strictly decreasing]] on the [[Definition:Closed Real Interval|interval]] $\closedint 0 \pi$
:$(2): \quad$ [[Definition:Strictly Increasing Real Function|strictly increasing]] on the [[Definition:Closed Real Inte... | {{improve|Extract those results into separate pages and reference them. Referencing the "discussion" is unsound because the proof can be changed and that "discussion" removed.}}
From the discussion of [[Real Cosine Function is Periodic]], we know that:
:$\cos x \ge 0$ on the [[Definition:Closed Real Interval|closed in... | Shape of Cosine Function | https://proofwiki.org/wiki/Shape_of_Cosine_Function | https://proofwiki.org/wiki/Shape_of_Cosine_Function | [
"Cosine Function"
] | [
"Definition:Cosine",
"Definition:Strictly Decreasing/Real Function",
"Definition:Real Interval/Closed",
"Definition:Strictly Increasing/Real Function",
"Definition:Real Interval/Closed",
"Definition:Concave Real Function",
"Definition:Real Interval/Closed",
"Definition:Convex Real Function",
"Defini... | [
"Sine and Cosine are Periodic on Reals/Cosine",
"Definition:Real Interval/Closed",
"Definition:Real Interval/Open",
"Sine and Cosine are Periodic on Reals/Cosine",
"Definition:Real Interval/Closed",
"Definition:Real Interval/Open",
"Derivative of Cosine Function",
"Derivative of Monotone Function",
... |
proofwiki-1741 | Tangent Function is Periodic on Reals | The real tangent function is periodic with period $\pi$.
This can be written:
:$\tan x = \map \tan {x \bmod \pi}$
where $x \bmod \pi$ denotes the modulo operation. | {{begin-eqn}}
{{eqn | l = \map \tan {x + \pi}
| r = \frac {\map \sin {x + \pi} } {\map \cos {x + \pi} }
| c = {{Defof|Real Tangent Function}}
}}
{{eqn | r = \frac {-\sin x} {-\cos x}
| c = Sine and Cosine are Periodic on Reals
}}
{{eqn | r = \tan x
| c =
}}
{{end-eqn}}
From Derivative of Tangen... | The [[Definition:Real Tangent Function|real tangent function]] is [[Definition:Periodic Real Function|periodic]] with [[Definition:Period of Periodic Real Function|period $\pi$]].
This can be written:
:$\tan x = \map \tan {x \bmod \pi}$
where $x \bmod \pi$ denotes the [[Definition:Modulo Operation|modulo operation]]. | {{begin-eqn}}
{{eqn | l = \map \tan {x + \pi}
| r = \frac {\map \sin {x + \pi} } {\map \cos {x + \pi} }
| c = {{Defof|Real Tangent Function}}
}}
{{eqn | r = \frac {-\sin x} {-\cos x}
| c = [[Sine and Cosine are Periodic on Reals]]
}}
{{eqn | r = \tan x
| c =
}}
{{end-eqn}}
From [[Derivative of... | Tangent Function is Periodic on Reals | https://proofwiki.org/wiki/Tangent_Function_is_Periodic_on_Reals | https://proofwiki.org/wiki/Tangent_Function_is_Periodic_on_Reals | [
"Tangent Function"
] | [
"Definition:Tangent Function/Real",
"Definition:Periodic Function/Real",
"Definition:Periodic Real Function/Period",
"Definition:Modulo Operation"
] | [
"Sine and Cosine are Periodic on Reals",
"Derivative of Tangent Function",
"Shape of Cosine Function",
"Definition:Real Interval/Open",
"Derivative of Monotone Function",
"Definition:Strictly Increasing/Real Function"
] |
proofwiki-1742 | Derivative of Tangent Function | :$\map {\dfrac \d {\d x} } {\tan x} = \sec^2 x = \dfrac 1 {\cos^2 x}$
when $\cos x \ne 0$. | From the definition of the tangent function:
:$\tan x = \dfrac {\sin x} {\cos x}$
From Derivative of Sine Function:
:$\map {\dfrac \d {\d x} } {\sin x} = \cos x$
From Derivative of Cosine Function:
:$\map {\dfrac \d {\d x} } {\cos x} = -\sin x$
Then:
{{begin-eqn}}
{{eqn | l = \map {\dfrac \d {\d x} } {\tan x}
| r... | :$\map {\dfrac \d {\d x} } {\tan x} = \sec^2 x = \dfrac 1 {\cos^2 x}$
when $\cos x \ne 0$. | From the definition of the [[Definition:Tangent Function|tangent]] function:
:$\tan x = \dfrac {\sin x} {\cos x}$
From [[Derivative of Sine Function]]:
:$\map {\dfrac \d {\d x} } {\sin x} = \cos x$
From [[Derivative of Cosine Function]]:
:$\map {\dfrac \d {\d x} } {\cos x} = -\sin x$
Then:
{{begin-eqn}}
{{eqn | l ... | Derivative of Tangent Function/Proof 1 | https://proofwiki.org/wiki/Derivative_of_Tangent_Function | https://proofwiki.org/wiki/Derivative_of_Tangent_Function/Proof_1 | [
"Derivative of Tangent Function",
"Tangent Function"
] | [] | [
"Definition:Tangent Function",
"Derivative of Sine Function",
"Derivative of Cosine Function",
"Quotient Rule for Derivatives",
"Sum of Squares of Sine and Cosine",
"Secant is Reciprocal of Cosine"
] |
proofwiki-1743 | Derivative of Tangent Function | :$\map {\dfrac \d {\d x} } {\tan x} = \sec^2 x = \dfrac 1 {\cos^2 x}$
when $\cos x \ne 0$. | {{begin-eqn}}
{{eqn | l = \map {\frac \d {\d x} } {\tan x}
| r = \lim_{h \mathop \to 0} \frac {\map \tan {x + h} - \tan x} h
| c = {{Defof|Derivative of Real Function at Point}}
}}
{{eqn | r = \lim_{h \mathop \to 0} \frac {\frac {\tan x + \tan h} {1 - \tan x \tan h} - \tan x} h
| c = Tangent of Sum
}}... | :$\map {\dfrac \d {\d x} } {\tan x} = \sec^2 x = \dfrac 1 {\cos^2 x}$
when $\cos x \ne 0$. | {{begin-eqn}}
{{eqn | l = \map {\frac \d {\d x} } {\tan x}
| r = \lim_{h \mathop \to 0} \frac {\map \tan {x + h} - \tan x} h
| c = {{Defof|Derivative of Real Function at Point}}
}}
{{eqn | r = \lim_{h \mathop \to 0} \frac {\frac {\tan x + \tan h} {1 - \tan x \tan h} - \tan x} h
| c = [[Tangent of Sum]... | Derivative of Tangent Function/Proof 2 | https://proofwiki.org/wiki/Derivative_of_Tangent_Function | https://proofwiki.org/wiki/Derivative_of_Tangent_Function/Proof_2 | [
"Derivative of Tangent Function",
"Tangent Function"
] | [] | [
"Tangent of Sum",
"Combination Theorem for Limits of Functions/Real/Product Rule",
"Limit of Tan X over X at Zero",
"Tangent of Zero",
"Sum of Squares of Sine and Cosine/Corollary 1",
"Secant is Reciprocal of Cosine"
] |
proofwiki-1744 | Stirling's Formula | The factorial function can be approximated by the formula:
:$n! \sim \sqrt {2 \pi n} \paren {\dfrac n e}^n$
where $\sim$ denotes asymptotically equal. | Let $a_n = \dfrac {n!} {\sqrt {2 n} \paren {\frac n e}^n}$.
=== Part 1 ===
It will be shown that:
:$\ds \lim_{n \mathop \to \infty} a_n = a$
for some constant $a$.
This will imply that:
:$\ds \lim_{n \mathop \to \infty} \frac {n!} {a \sqrt{2 n} \paren {\frac n e}^n} = 1$
By applying Power Series Expansion for $\map \ln... | The [[Definition:Factorial|factorial]] function can be approximated by the formula:
:$n! \sim \sqrt {2 \pi n} \paren {\dfrac n e}^n$
where $\sim$ denotes [[Definition:Asymptotically Equal Sequences|asymptotically equal]]. | Let $a_n = \dfrac {n!} {\sqrt {2 n} \paren {\frac n e}^n}$.
=== Part 1 ===
It will be shown that:
:$\ds \lim_{n \mathop \to \infty} a_n = a$
for some [[Definition:Constant|constant]] $a$.
This will imply that:
:$\ds \lim_{n \mathop \to \infty} \frac {n!} {a \sqrt{2 n} \paren {\frac n e}^n} = 1$
By applying [[Pow... | Stirling's Formula/Proof 1 | https://proofwiki.org/wiki/Stirling's_Formula | https://proofwiki.org/wiki/Stirling's_Formula/Proof_1 | [
"Stirling's Formula",
"Factorials",
"Asymptotics",
"Special Functions"
] | [
"Definition:Factorial",
"Definition:Asymptotic Equality/Sequences"
] | [
"Definition:Constant",
"Power Series Expansion for Logarithm of 1 + x",
"Difference of Logarithms",
"Sum of Logarithms",
"Difference of Logarithms",
"Power Series Expansion for Logarithm of 1 + x",
"Power Series Expansion for Logarithm of 1 - x",
"Definition:Even Integer",
"Definition:Term of Expres... |
proofwiki-1745 | Stirling's Formula | The factorial function can be approximated by the formula:
:$n! \sim \sqrt {2 \pi n} \paren {\dfrac n e}^n$
where $\sim$ denotes asymptotically equal. | Consider the sequence $\sequence {d_n}$ defined as:
:$d_n = \map \ln {n!} - \paren {n + \dfrac 1 2} \ln n + n$
From Lemma 2 it is seen that $\sequence {d_n}$ is a decreasing sequence.
From Lemma 3 it is seen that the sequence:
:$\sequence {d_n - \dfrac 1 {12 n} }$
is increasing.
In particular:
:$\forall n \in \N_{>0}: ... | The [[Definition:Factorial|factorial]] function can be approximated by the formula:
:$n! \sim \sqrt {2 \pi n} \paren {\dfrac n e}^n$
where $\sim$ denotes [[Definition:Asymptotically Equal Sequences|asymptotically equal]]. | Consider the [[Definition:Sequence|sequence]] $\sequence {d_n}$ defined as:
:$d_n = \map \ln {n!} - \paren {n + \dfrac 1 2} \ln n + n$
From [[Stirling's Formula/Proof 2/Lemma 2|Lemma 2]] it is seen that $\sequence {d_n}$ is a [[Definition:Decreasing Real Sequence|decreasing sequence]].
From [[Stirling's Formula/Proo... | Stirling's Formula/Proof 2 | https://proofwiki.org/wiki/Stirling's_Formula | https://proofwiki.org/wiki/Stirling's_Formula/Proof_2 | [
"Stirling's Formula",
"Factorials",
"Asymptotics",
"Special Functions"
] | [
"Definition:Factorial",
"Definition:Asymptotic Equality/Sequences"
] | [
"Definition:Sequence",
"Stirling's Formula/Proof 2/Lemma 2",
"Definition:Decreasing/Sequence/Real Sequence",
"Stirling's Formula/Proof 2/Lemma 3",
"Definition:Sequence",
"Definition:Increasing/Sequence/Real Sequence",
"Definition:Bounded Below Sequence/Real",
"Monotone Convergence Theorem (Real Analys... |
proofwiki-1746 | Stirling's Formula | The factorial function can be approximated by the formula:
:$n! \sim \sqrt {2 \pi n} \paren {\dfrac n e}^n$
where $\sim$ denotes asymptotically equal. | From Poisson Distribution Approximated by Normal Distribution, we have:
Let $X$ be a discrete random variable which has the Poisson distribution $\Poisson \lambda$.
Then for large $\lambda$:
:$\Poisson \lambda \approx \Gaussian \lambda \lambda$
where $\Gaussian \lambda \lambda$ denotes the normal distribution.
Since th... | The [[Definition:Factorial|factorial]] function can be approximated by the formula:
:$n! \sim \sqrt {2 \pi n} \paren {\dfrac n e}^n$
where $\sim$ denotes [[Definition:Asymptotically Equal Sequences|asymptotically equal]]. | From [[Poisson Distribution Approximated by Normal Distribution]], we have:
Let $X$ be a [[Definition:Discrete Random Variable|discrete random variable]] which has the [[Definition:Poisson Distribution|Poisson distribution $\Poisson \lambda$]].
Then for large $\lambda$:
:$\Poisson \lambda \approx \Gaussian \lambda \l... | Stirling's Formula/Proof 3 | https://proofwiki.org/wiki/Stirling's_Formula | https://proofwiki.org/wiki/Stirling's_Formula/Proof_3 | [
"Stirling's Formula",
"Factorials",
"Asymptotics",
"Special Functions"
] | [
"Definition:Factorial",
"Definition:Asymptotic Equality/Sequences"
] | [
"Poisson Distribution Approximated by Normal Distribution",
"Definition:Random Variable/Discrete",
"Definition:Poisson Distribution",
"Definition:Normal Distribution",
"Definition:Poisson Distribution",
"Definition:Family of Probability Distributions/Parameter",
"Definition:Normal Distribution",
"Defi... |
proofwiki-1747 | Stirling's Formula | The factorial function can be approximated by the formula:
:$n! \sim \sqrt {2 \pi n} \paren {\dfrac n e}^n$
where $\sim$ denotes asymptotically equal. | Let:
:$\ds \map f n := \dfrac {n!} {\sqrt {2 \pi n} n^n e^{-n} } - \paren {1 + \frac 1 {12 n} }$
We need to show:
:$\ds \map f n = \map \OO {\dfrac 1 {n^2} }$
Recall Limit of Error in Stirling's Formula:
:$e^{1 / \paren {12 n + 1} } \le \dfrac {n!} {\sqrt {2 \pi n} n^n e^{-n} } \le e^{1 / 12 n}$
Furthermore, observe:
{... | The [[Definition:Factorial|factorial]] function can be approximated by the formula:
:$n! \sim \sqrt {2 \pi n} \paren {\dfrac n e}^n$
where $\sim$ denotes [[Definition:Asymptotically Equal Sequences|asymptotically equal]]. | Let:
:$\ds \map f n := \dfrac {n!} {\sqrt {2 \pi n} n^n e^{-n} } - \paren {1 + \frac 1 {12 n} }$
We need to show:
:$\ds \map f n = \map \OO {\dfrac 1 {n^2} }$
Recall [[Limit of Error in Stirling's Formula]]:
:$e^{1 / \paren {12 n + 1} } \le \dfrac {n!} {\sqrt {2 \pi n} n^n e^{-n} } \le e^{1 / 12 n}$
Furthermore, ob... | Stirling's Formula/Refinement/Proof 1 | https://proofwiki.org/wiki/Stirling's_Formula | https://proofwiki.org/wiki/Stirling's_Formula/Refinement/Proof_1 | [
"Stirling's Formula",
"Factorials",
"Asymptotics",
"Special Functions"
] | [
"Definition:Factorial",
"Definition:Asymptotic Equality/Sequences"
] | [
"Limit of Error in Stirling's Formula",
"Exponential of x not less than 1+x",
"Taylor's Theorem",
"Exponential is Strictly Increasing",
"Definition:Euler's Number/Decimal Expansion"
] |
proofwiki-1748 | Stirling's Formula | The factorial function can be approximated by the formula:
:$n! \sim \sqrt {2 \pi n} \paren {\dfrac n e}^n$
where $\sim$ denotes asymptotically equal. | Let $z\in \R_{>0}$ and $n \in \N_{\ge 0}$
Let $\ds c_n = \ln \map \Gamma {z + n}$
We begin by observing:
{{begin-eqn}}
{{eqn | l = \map \Gamma {z + n}
| r = \map \Gamma {z + 1} \times \paren {z + 1} \times \paren {z + 2} \times \cdots \times \paren {z + n - 1}
| c = Gamma Difference Equation
}}
{{eqn | ll =... | The [[Definition:Factorial|factorial]] function can be approximated by the formula:
:$n! \sim \sqrt {2 \pi n} \paren {\dfrac n e}^n$
where $\sim$ denotes [[Definition:Asymptotically Equal Sequences|asymptotically equal]]. | Let $z\in \R_{>0}$ and $n \in \N_{\ge 0}$
Let $\ds c_n = \ln \map \Gamma {z + n}$
We begin by observing:
{{begin-eqn}}
{{eqn | l = \map \Gamma {z + n}
| r = \map \Gamma {z + 1} \times \paren {z + 1} \times \paren {z + 2} \times \cdots \times \paren {z + n - 1}
| c = [[Gamma Difference Equation]]
}}
{{eqn ... | Stirling's Formula/Refinement/Proof 2 | https://proofwiki.org/wiki/Stirling's_Formula | https://proofwiki.org/wiki/Stirling's_Formula/Refinement/Proof_2 | [
"Stirling's Formula",
"Factorials",
"Asymptotics",
"Special Functions"
] | [
"Definition:Factorial",
"Definition:Asymptotic Equality/Sequences"
] | [
"Gamma Difference Equation",
"Sum of Logarithms",
"Definition:Derivative",
"Definition:Finite Difference Operator",
"Primitive of Logarithm of x",
"Definition:Derivative",
"Definition:Finite Difference Operator",
"Primitive of Logarithm of x",
"Sum of Logarithms",
"Power Series Expansion for Logar... |
proofwiki-1749 | Stirling's Formula | The factorial function can be approximated by the formula:
:$n! \sim \sqrt {2 \pi n} \paren {\dfrac n e}^n$
where $\sim$ denotes asymptotically equal. | From Limit of Error in Stirling's Formula, we have:
:$e^{1 / \paren {12 n + 1} } < \dfrac {n!} {\sqrt {2 \pi n} n^n e^{-n} } < e^{1 / 12 n}$
We also have:
{{begin-eqn}}
{{eqn | l = e^{1 / 12 n}
| r = 1 + \frac 1 {12 n} + \frac 1 {2!} \paren {\frac 1 {12 n} }^2 + \frac 1 {3!} \paren {\frac 1 {12 n} }^3 + \cdots
... | The [[Definition:Factorial|factorial]] function can be approximated by the formula:
:$n! \sim \sqrt {2 \pi n} \paren {\dfrac n e}^n$
where $\sim$ denotes [[Definition:Asymptotically Equal Sequences|asymptotically equal]]. | From [[Limit of Error in Stirling's Formula]], we have:
:$e^{1 / \paren {12 n + 1} } < \dfrac {n!} {\sqrt {2 \pi n} n^n e^{-n} } < e^{1 / 12 n}$
We also have:
{{begin-eqn}}
{{eqn | l = e^{1 / 12 n}
| r = 1 + \frac 1 {12 n} + \frac 1 {2!} \paren {\frac 1 {12 n} }^2 + \frac 1 {3!} \paren {\frac 1 {12 n} }^3 + \cd... | Stirling's Formula/Refinement/Proof 3 | https://proofwiki.org/wiki/Stirling's_Formula | https://proofwiki.org/wiki/Stirling's_Formula/Refinement/Proof_3 | [
"Stirling's Formula",
"Factorials",
"Asymptotics",
"Special Functions"
] | [
"Definition:Factorial",
"Definition:Asymptotic Equality/Sequences"
] | [
"Limit of Error in Stirling's Formula",
"Power Series Expansion for Exponential Function",
"Power Series Expansion for Exponential Function",
"Definition:Sufficiently Large",
"Limit of Error in Stirling's Formula"
] |
proofwiki-1750 | Shape of Sine Function | The sine function is:
:$(1): \quad$ strictly increasing on the interval $\closedint {-\dfrac \pi 2} {\dfrac \pi 2}$
:$(2): \quad$ strictly decreasing on the interval $\closedint {\dfrac \pi 2} {\dfrac {3 \pi} 2}$
:$(3): \quad$ concave on the interval $\closedint 0 \pi$
:$(4): \quad$ convex on the interval $\closedint \... | From the discussion of Real Sine Function is Periodic, we have that:
:$\sin \paren {x + \dfrac \pi 2} = \cos x$
The result then follows directly from the Shape of Cosine Function. | The [[Definition:Sine|sine]] function is:
:$(1): \quad$ [[Definition:Strictly Increasing Real Function|strictly increasing]] on the [[Definition:Closed Real Interval|interval]] $\closedint {-\dfrac \pi 2} {\dfrac \pi 2}$
:$(2): \quad$ [[Definition:Strictly Decreasing Real Function|strictly decreasing]] on the [[Defin... | From the discussion of [[Real Sine Function is Periodic]], we have that:
:$\sin \paren {x + \dfrac \pi 2} = \cos x$
The result then follows directly from the [[Shape of Cosine Function]]. | Shape of Sine Function | https://proofwiki.org/wiki/Shape_of_Sine_Function | https://proofwiki.org/wiki/Shape_of_Sine_Function | [
"Sine Function"
] | [
"Definition:Sine",
"Definition:Strictly Increasing/Real Function",
"Definition:Real Interval/Closed",
"Definition:Strictly Decreasing/Real Function",
"Definition:Real Interval/Closed",
"Definition:Concave Real Function",
"Definition:Real Interval/Closed",
"Definition:Convex Real Function",
"Definiti... | [
"Sine and Cosine are Periodic on Reals/Sine",
"Shape of Cosine Function"
] |
proofwiki-1751 | Derivative of Secant Function | :$\map {\dfrac \d {\d x} } {\sec x} = \sec x \tan x$
where $\cos x \ne 0$. | {{begin-eqn}}
{{eqn | l = \map {\dfrac \d {\d x} } {\sec x}
| r = \map {\dfrac \d {\d x} } {\dfrac 1 {\cos x} }
| c = {{Defof|Real Secant Function}}
}}
{{eqn | r = \dfrac {\cos x \map {\frac \d {\d x} } 1 - 1 \map {\frac \d {\d x} } {\cos x} } {\cos^2 x}
| c = Quotient Rule for Derivatives
}}
{{eqn | ... | :$\map {\dfrac \d {\d x} } {\sec x} = \sec x \tan x$
where $\cos x \ne 0$. | {{begin-eqn}}
{{eqn | l = \map {\dfrac \d {\d x} } {\sec x}
| r = \map {\dfrac \d {\d x} } {\dfrac 1 {\cos x} }
| c = {{Defof|Real Secant Function}}
}}
{{eqn | r = \dfrac {\cos x \map {\frac \d {\d x} } 1 - 1 \map {\frac \d {\d x} } {\cos x} } {\cos^2 x}
| c = [[Quotient Rule for Derivatives]]
}}
{{eq... | Derivative of Secant Function/Proof 2 | https://proofwiki.org/wiki/Derivative_of_Secant_Function | https://proofwiki.org/wiki/Derivative_of_Secant_Function/Proof_2 | [
"Derivative of Secant Function",
"Derivatives of Trigonometric Functions",
"Secant Function"
] | [] | [
"Quotient Rule for Derivatives",
"Derivative of Cosine Function",
"Derivative of Constant"
] |
proofwiki-1752 | Derivative of Cosecant Function | :$\map {\dfrac \d {\d x} } {\csc x} = -\csc x \cot x$
where $\sin x \ne 0$. | * {{BookReference|A Note Book in Pure Mathematics|1953|L. Harwood Clarke|prev = Derivative of Tangent Function/Proof 1|next = Derivative of Secant Function/Proof 2}}: $\text {II}$. Calculus: Differentiation: Quotient
Category:Derivative of Cosecant Function
f16wucwytsunbruw1jlda17nropk15w | :$\map {\dfrac \d {\d x} } {\csc x} = -\csc x \cot x$
where $\sin x \ne 0$. | * {{BookReference|A Note Book in Pure Mathematics|1953|L. Harwood Clarke|prev = Derivative of Tangent Function/Proof 1|next = Derivative of Secant Function/Proof 2}}: $\text {II}$. Calculus: Differentiation: Quotient
[[Category:Derivative of Cosecant Function]]
f16wucwytsunbruw1jlda17nropk15w | Derivative of Cosecant Function/Proof 2 | https://proofwiki.org/wiki/Derivative_of_Cosecant_Function | https://proofwiki.org/wiki/Derivative_of_Cosecant_Function/Proof_2 | [
"Derivative of Cosecant Function",
"Derivatives of Trigonometric Functions",
"Cosecant Function"
] | [] | [
"Category:Derivative of Cosecant Function"
] |
proofwiki-1753 | Shape of Tangent Function | The nature of the tangent function on the set of real numbers $\R$ is as follows:
:$\tan x$ is continuous and strictly increasing on the interval $\openint {-\dfrac \pi 2} {\dfrac \pi 2}$
:$\tan x \to +\infty$ as $x \to \dfrac \pi 2 ^-$
:$\tan x \to -\infty$ as $x \to -\dfrac \pi 2 ^+$
:$\tan x$ is not defined on $\for... | $\tan x$ is continuous and strictly increasing on $\openint {-\dfrac \pi 2} {\dfrac \pi 2}$:
Continuity follows from the Quotient Rule for Continuous Real Functions:
:$(1): \quad$ Both $\sin x$ and $\cos x$ are continuous on $\openint {-\dfrac \pi 2} {\dfrac \pi 2}$ from Real Sine Function is Continuous and Cosine Func... | The nature of the [[Definition:Real Tangent Function|tangent function]] on the set of [[Definition:Real Number|real numbers]] $\R$ is as follows:
:$\tan x$ is [[Definition:Continuous on Interval|continuous]] and [[Definition:Strictly Increasing Real Function|strictly increasing]] on the [[Definition:Open Real Interval... | $\tan x$ is [[Definition:Continuous on Interval|continuous]] and [[Definition:Strictly Increasing Real Function|strictly increasing]] on $\openint {-\dfrac \pi 2} {\dfrac \pi 2}$:
Continuity follows from the [[Quotient Rule for Continuous Real Functions]]:
:$(1): \quad$ Both $\sin x$ and $\cos x$ are [[Definition:Con... | Shape of Tangent Function | https://proofwiki.org/wiki/Shape_of_Tangent_Function | https://proofwiki.org/wiki/Shape_of_Tangent_Function | [
"Tangent Function"
] | [
"Definition:Tangent Function/Real",
"Definition:Real Number",
"Definition:Continuous Real Function/Interval",
"Definition:Strictly Increasing/Real Function",
"Definition:Real Interval/Open",
"Definition:Discontinuous"
] | [
"Definition:Continuous Real Function/Interval",
"Definition:Strictly Increasing/Real Function",
"Combination Theorem for Continuous Functions/Real/Quotient Rule",
"Definition:Continuous Real Function/Interval",
"Real Sine Function is Continuous",
"Cosine Function is Continuous",
"Definition:Strictly Inc... |
proofwiki-1754 | Derivative of Arcsine Function | :$\dfrac {\map \d {\arcsin x} } {\d x} = \dfrac 1 {\sqrt {1 - x^2} }$ | Let $y = \arcsin x$ where $-1 < x < 1$.
Then:
{{begin-eqn}}
{{eqn | l = x
| r = \sin y
| c =
}}
{{eqn | ll= \leadsto
| l = \dfrac {\d x} {\d y}
| r = \cos y
| c = Derivative of Sine Function
}}
{{end-eqn}}
Then:
{{begin-eqn}}
{{eqn | l = \cos^2 y + \sin^2 y
| r = 1
| c = Sum o... | :$\dfrac {\map \d {\arcsin x} } {\d x} = \dfrac 1 {\sqrt {1 - x^2} }$ | Let $y = \arcsin x$ where $-1 < x < 1$.
Then:
{{begin-eqn}}
{{eqn | l = x
| r = \sin y
| c =
}}
{{eqn | ll= \leadsto
| l = \dfrac {\d x} {\d y}
| r = \cos y
| c = [[Derivative of Sine Function]]
}}
{{end-eqn}}
Then:
{{begin-eqn}}
{{eqn | l = \cos^2 y + \sin^2 y
| r = 1
|... | Derivative of Arcsine Function/Proof | https://proofwiki.org/wiki/Derivative_of_Arcsine_Function | https://proofwiki.org/wiki/Derivative_of_Arcsine_Function/Proof | [
"Derivative of Arcsine Function",
"Derivatives of Inverse Trigonometric Functions",
"Arcsine Function"
] | [] | [
"Derivative of Sine Function",
"Sum of Squares of Sine and Cosine",
"Definition:Image (Set Theory)/Mapping/Mapping"
] |
proofwiki-1755 | Derivative of Arccosine Function | :$\map {D_x} {\arccos x} = \dfrac {-1} {\sqrt {1 - x^2} }$ | Let $y = \arccos x$ where $-1 < x < 1$.
Then:
:$x = \cos y$
Then from Derivative of Cosine Function:
:$\dfrac {\d x} {\d y} = -\sin y$
Hence from Derivative of Inverse Function:
:$\dfrac {\d y} {\d x} = \dfrac {-1} {\sin y}$
From Sum of Squares of Sine and Cosine, we have:
:$\cos^2 y + \sin^2 y = 1 \implies \sin y = \p... | :$\map {D_x} {\arccos x} = \dfrac {-1} {\sqrt {1 - x^2} }$ | Let $y = \arccos x$ where $-1 < x < 1$.
Then:
:$x = \cos y$
Then from [[Derivative of Cosine Function]]:
:$\dfrac {\d x} {\d y} = -\sin y$
Hence from [[Derivative of Inverse Function]]:
:$\dfrac {\d y} {\d x} = \dfrac {-1} {\sin y}$
From [[Sum of Squares of Sine and Cosine]], we have:
:$\cos^2 y + \sin^2 y = 1 \imp... | Derivative of Arccosine Function | https://proofwiki.org/wiki/Derivative_of_Arccosine_Function | https://proofwiki.org/wiki/Derivative_of_Arccosine_Function | [
"Derivatives of Inverse Trigonometric Functions",
"Arccosine Function"
] | [] | [
"Derivative of Cosine Function",
"Derivative of Inverse Function",
"Sum of Squares of Sine and Cosine"
] |
proofwiki-1756 | Sum of Arcsine and Arccosine | Let $x \in \R$ be a real number such that $-1 \le x \le 1$.
Then:
:$\arcsin x + \arccos x = \dfrac \pi 2$
where $\arcsin$ and $\arccos$ denote arcsine and arccosine respectively. | Let $y \in \R$ such that:
:$\exists x \in \closedint {-1} 1: x = \map \cos {y + \dfrac \pi 2}$
Then:
{{begin-eqn}}
{{eqn | l = x
| r = \map \cos {y + \frac \pi 2}
| c =
}}
{{eqn | r = -\sin y
| c = Cosine of Angle plus Right Angle
}}
{{eqn | r = \map \sin {-y}
| c = Sine Function is Odd
}}
{{en... | Let $x \in \R$ be a [[Definition:Real Number|real number]] such that $-1 \le x \le 1$.
Then:
:$\arcsin x + \arccos x = \dfrac \pi 2$
where $\arcsin$ and $\arccos$ denote [[Definition:Real Arcsine|arcsine]] and [[Definition:Real Arccosine|arccosine]] respectively. | Let $y \in \R$ such that:
:$\exists x \in \closedint {-1} 1: x = \map \cos {y + \dfrac \pi 2}$
Then:
{{begin-eqn}}
{{eqn | l = x
| r = \map \cos {y + \frac \pi 2}
| c =
}}
{{eqn | r = -\sin y
| c = [[Cosine of Angle plus Right Angle]]
}}
{{eqn | r = \map \sin {-y}
| c = [[Sine Function is Odd]... | Sum of Arcsine and Arccosine/Proof 1 | https://proofwiki.org/wiki/Sum_of_Arcsine_and_Arccosine | https://proofwiki.org/wiki/Sum_of_Arcsine_and_Arccosine/Proof_1 | [
"Sum of Arcsine and Arccosine",
"Arcsine Function",
"Arccosine Function"
] | [
"Definition:Real Number",
"Definition:Inverse Sine/Real/Arcsine",
"Definition:Inverse Cosine/Real/Arccosine"
] | [
"Cosine of Angle plus Right Angle",
"Sine Function is Odd"
] |
proofwiki-1757 | Sum of Arcsine and Arccosine | Let $x \in \R$ be a real number such that $-1 \le x \le 1$.
Then:
:$\arcsin x + \arccos x = \dfrac \pi 2$
where $\arcsin$ and $\arccos$ denote arcsine and arccosine respectively. | :400px
Consider the diagram above.
{{begin-eqn}}
{{eqn | l = \alpha
| r = \arcsin A
| c =
}}
{{eqn | l = \beta
| r = \arccos A
| c =
}}
{{eqn | l = \alpha + \beta
| r = \frac \pi 2
| c = Sum of Angles of Triangle equals Two Right Angles
}}
{{end-eqn}}
{{qed}} | Let $x \in \R$ be a [[Definition:Real Number|real number]] such that $-1 \le x \le 1$.
Then:
:$\arcsin x + \arccos x = \dfrac \pi 2$
where $\arcsin$ and $\arccos$ denote [[Definition:Real Arcsine|arcsine]] and [[Definition:Real Arccosine|arccosine]] respectively. | :[[File:Sum_of_arcsin_and_arc_cos.png|400px]]
Consider the diagram above.
{{begin-eqn}}
{{eqn | l = \alpha
| r = \arcsin A
| c =
}}
{{eqn | l = \beta
| r = \arccos A
| c =
}}
{{eqn | l = \alpha + \beta
| r = \frac \pi 2
| c = [[Sum of Angles of Triangle equals Two Right Angles]]
... | Sum of Arcsine and Arccosine/Proof 2 | https://proofwiki.org/wiki/Sum_of_Arcsine_and_Arccosine | https://proofwiki.org/wiki/Sum_of_Arcsine_and_Arccosine/Proof_2 | [
"Sum of Arcsine and Arccosine",
"Arcsine Function",
"Arccosine Function"
] | [
"Definition:Real Number",
"Definition:Inverse Sine/Real/Arcsine",
"Definition:Inverse Cosine/Real/Arccosine"
] | [
"File:Sum_of_arcsin_and_arc_cos.png",
"Sum of Angles of Triangle equals Two Right Angles"
] |
proofwiki-1758 | Derivative of Arctangent Function | :$\dfrac {\map \d {\arctan x} } {\d x} = \dfrac 1 {1 + x^2}$ | {{begin-eqn}}
{{eqn | l = y
| r = \arctan x
| c =
}}
{{eqn | ll= \leadsto
| l = x
| r = \tan y
| c = {{Defof|Real Arctangent}}
}}
{{eqn | ll= \leadsto
| l = \frac {\d x} {\d y}
| r = \sec^2 y
| c = Derivative of Tangent Function
}}
{{eqn | r = 1 + \tan^2 y
| c = Di... | :$\dfrac {\map \d {\arctan x} } {\d x} = \dfrac 1 {1 + x^2}$ | {{begin-eqn}}
{{eqn | l = y
| r = \arctan x
| c =
}}
{{eqn | ll= \leadsto
| l = x
| r = \tan y
| c = {{Defof|Real Arctangent}}
}}
{{eqn | ll= \leadsto
| l = \frac {\d x} {\d y}
| r = \sec^2 y
| c = [[Derivative of Tangent Function]]
}}
{{eqn | r = 1 + \tan^2 y
| c ... | Derivative of Arctangent Function/Proof 1 | https://proofwiki.org/wiki/Derivative_of_Arctangent_Function | https://proofwiki.org/wiki/Derivative_of_Arctangent_Function/Proof_1 | [
"Derivative of Arctangent Function",
"Arctangent Function",
"Derivatives of Inverse Trigonometric Functions"
] | [] | [
"Derivative of Tangent Function",
"Sum of Squares of Sine and Cosine/Corollary 1",
"Derivative of Inverse Function"
] |
proofwiki-1759 | Derivative of Arctangent Function | :$\dfrac {\map \d {\arctan x} } {\d x} = \dfrac 1 {1 + x^2}$ | {{begin-eqn}}
{{eqn | l = \frac {\map \d {\arctan x} } {\d x}
| r = \lim_{h \mathop \to 0} \frac {\map \arctan {x + h} - \arctan x} h
| c = {{Defof|Derivative of Real Function at Point}}
}}
{{eqn | r = \lim_{h \mathop \to 0} \frac {\map \arctan {x + h} + \map \arctan {-x} } h
| c = Arctangent Funct... | :$\dfrac {\map \d {\arctan x} } {\d x} = \dfrac 1 {1 + x^2}$ | {{begin-eqn}}
{{eqn | l = \frac {\map \d {\arctan x} } {\d x}
| r = \lim_{h \mathop \to 0} \frac {\map \arctan {x + h} - \arctan x} h
| c = {{Defof|Derivative of Real Function at Point}}
}}
{{eqn | r = \lim_{h \mathop \to 0} \frac {\map \arctan {x + h} + \map \arctan {-x} } h
| c = [[Arctangent Fun... | Derivative of Arctangent Function/Proof 2 | https://proofwiki.org/wiki/Derivative_of_Arctangent_Function | https://proofwiki.org/wiki/Derivative_of_Arctangent_Function/Proof_2 | [
"Derivative of Arctangent Function",
"Arctangent Function",
"Derivatives of Inverse Trigonometric Functions"
] | [] | [
"Inverse Tangent is Odd Function",
"Sum of Arctangents"
] |
proofwiki-1760 | Complex Numbers form Field | Consider the algebraic structure $\struct {\C, +, \times}$, where:
:$\C$ is the set of all complex numbers
:$+$ is the operation of complex addition
:$\times$ is the operation of complex multiplication
Then $\struct {\C, +, \times}$ forms a field. | From Complex Numbers under Addition form Infinite Abelian Group, we have that $\struct {\C, +}$ forms an abelian group.
From Non-Zero Complex Numbers under Multiplication form Infinite Abelian Group, we have that $\struct {\C_{\ne 0}, \times}$ forms an abelian group.
Finally, we have that Complex Multiplication Distrib... | Consider the [[Definition:Algebraic Structure with Two Operations|algebraic structure]] $\struct {\C, +, \times}$, where:
:$\C$ is the set of all [[Definition:Complex Number|complex numbers]]
:$+$ is the operation of [[Definition:Complex Addition|complex addition]]
:$\times$ is the operation of [[Definition:Complex Mu... | From [[Complex Numbers under Addition form Infinite Abelian Group]], we have that $\struct {\C, +}$ forms an [[Definition:Abelian Group|abelian group]].
From [[Non-Zero Complex Numbers under Multiplication form Infinite Abelian Group]], we have that $\struct {\C_{\ne 0}, \times}$ forms an [[Definition:Abelian Group|ab... | Complex Numbers form Field | https://proofwiki.org/wiki/Complex_Numbers_form_Field | https://proofwiki.org/wiki/Complex_Numbers_form_Field | [
"Complex Numbers",
"Examples of Fields"
] | [
"Definition:Algebraic Structure/Two Operations",
"Definition:Complex Number",
"Definition:Addition/Complex Numbers",
"Definition:Multiplication/Complex Numbers",
"Definition:Field (Abstract Algebra)"
] | [
"Complex Numbers under Addition form Infinite Abelian Group",
"Definition:Abelian Group",
"Non-Zero Complex Numbers under Multiplication form Infinite Abelian Group",
"Definition:Abelian Group",
"Complex Multiplication Distributes over Addition",
"Definition:Field (Abstract Algebra)"
] |
proofwiki-1761 | Complex Multiplication is Commutative | The operation of multiplication on the set of complex numbers $\C$ is commutative:
:$\forall z_1, z_2 \in \C: z_1 z_2 = z_2 z_1$ | From the definition of complex numbers, we define the following:
{{begin-eqn}}
{{eqn | l = z
| o = :=
| r = \tuple {x_1, y_1}
}}
{{eqn | l = w
| o = :=
| r = \tuple {x_2, y_2}
}}
{{end-eqn}}
where $x_1, x_2, y_1, y_2 \in \R$.
Then:
{{begin-eqn}}
{{eqn | l = z_1 z_2
| r = \tuple {x_1, y_1} ... | The operation of [[Definition:Complex Multiplication|multiplication]] on the [[Definition:Set|set]] of [[Definition:Complex Number|complex numbers]] $\C$ is [[Definition:Commutative Operation|commutative]]:
:$\forall z_1, z_2 \in \C: z_1 z_2 = z_2 z_1$ | From the definition of [[Definition:Complex Number/Definition 2|complex numbers]], we define the following:
{{begin-eqn}}
{{eqn | l = z
| o = :=
| r = \tuple {x_1, y_1}
}}
{{eqn | l = w
| o = :=
| r = \tuple {x_2, y_2}
}}
{{end-eqn}}
where $x_1, x_2, y_1, y_2 \in \R$.
Then:
{{begin-eqn}}
{{... | Complex Multiplication is Commutative | https://proofwiki.org/wiki/Complex_Multiplication_is_Commutative | https://proofwiki.org/wiki/Complex_Multiplication_is_Commutative | [
"Complex Multiplication",
"Commutative Law of Multiplication",
"Examples of Commutative Operations"
] | [
"Definition:Multiplication/Complex Numbers",
"Definition:Set",
"Definition:Complex Number",
"Definition:Commutative/Operation"
] | [
"Definition:Complex Number/Definition 2",
"Real Multiplication is Commutative",
"Real Addition is Commutative"
] |
proofwiki-1762 | Complex Multiplication Distributes over Addition | The operation of multiplication on the set of complex numbers $\C$ is distributive over the operation of addition.
:$\forall z_1, z_2, z_3 \in \C:$
::$z_1 \paren {z_2 + z_3} = z_1 z_2 + z_1 z_3$
::$\paren {z_2 + z_3} z_1 = z_2 z_1 + z_3 z_1$ | From the definition of complex numbers, we define the following:
{{begin-eqn}}
{{eqn | l = z_1
| o = :=
| r = \tuple {x_1, y_1}
}}
{{eqn | l = z_2
| o = :=
| r = \tuple {x_2, y_2}
}}
{{eqn | l = z_3
| o = :=
| r = \tuple {x_3, y_3}
}}
{{end-eqn}}
where $x_1, x_2, x_3, y_1, y_2, y_3 \... | The operation of [[Definition:Complex Multiplication|multiplication]] on the [[Definition:Set|set]] of [[Definition:Complex Number|complex numbers]] $\C$ is [[Definition:Distributive Operation|distributive]] over the operation of [[Definition:Complex Addition|addition]].
:$\forall z_1, z_2, z_3 \in \C:$
::$z_1 \paren {... | From the definition of [[Definition:Complex Number/Definition 2|complex numbers]], we define the following:
{{begin-eqn}}
{{eqn | l = z_1
| o = :=
| r = \tuple {x_1, y_1}
}}
{{eqn | l = z_2
| o = :=
| r = \tuple {x_2, y_2}
}}
{{eqn | l = z_3
| o = :=
| r = \tuple {x_3, y_3}
}}
{{end... | Complex Multiplication Distributes over Addition | https://proofwiki.org/wiki/Complex_Multiplication_Distributes_over_Addition | https://proofwiki.org/wiki/Complex_Multiplication_Distributes_over_Addition | [
"Complex Multiplication",
"Complex Addition",
"Examples of Distributive Operations"
] | [
"Definition:Multiplication/Complex Numbers",
"Definition:Set",
"Definition:Complex Number",
"Definition:Distributive Operation",
"Definition:Addition/Complex Numbers"
] | [
"Definition:Complex Number/Definition 2",
"Real Multiplication Distributes over Addition",
"Real Addition is Commutative",
"Complex Multiplication is Commutative"
] |
proofwiki-1763 | Complex Modulus of Product of Complex Numbers | Let $z_1, z_2 \in \C$ be complex numbers.
Let $\cmod z$ be the modulus of $z$.
Then:
:$\cmod {z_1 z_2} = \cmod {z_1} \cdot \cmod {z_2}$ | Let $z_1 = x_1 + i y_1$ and $z_2 = x_2 + i y_2$, where $x_1, y_1, x_2, y_2 \in \R$.
{{begin-eqn}}
{{eqn | l = \cmod {z_1 z_2}
| r = \sqrt {\paren {x_1 x_2 - y_1 y_2}^2 + \paren {x_1 y_2 + x_2 y_1}^2}
| c = {{Defof|Complex Modulus}}, {{Defof|Complex Multiplication}}
}}
{{eqn | r = \sqrt {\paren {x_1^2 x_2^2 ... | Let $z_1, z_2 \in \C$ be [[Definition:Complex Number|complex numbers]].
Let $\cmod z$ be the [[Definition:Complex Modulus|modulus]] of $z$.
Then:
:$\cmod {z_1 z_2} = \cmod {z_1} \cdot \cmod {z_2}$ | Let $z_1 = x_1 + i y_1$ and $z_2 = x_2 + i y_2$, where $x_1, y_1, x_2, y_2 \in \R$.
{{begin-eqn}}
{{eqn | l = \cmod {z_1 z_2}
| r = \sqrt {\paren {x_1 x_2 - y_1 y_2}^2 + \paren {x_1 y_2 + x_2 y_1}^2}
| c = {{Defof|Complex Modulus}}, {{Defof|Complex Multiplication}}
}}
{{eqn | r = \sqrt {\paren {x_1^2 x_2^... | Complex Modulus of Product of Complex Numbers/Proof 1 | https://proofwiki.org/wiki/Complex_Modulus_of_Product_of_Complex_Numbers | https://proofwiki.org/wiki/Complex_Modulus_of_Product_of_Complex_Numbers/Proof_1 | [
"Complex Modulus",
"Complex Multiplication",
"Complex Modulus of Product of Complex Numbers"
] | [
"Definition:Complex Number",
"Definition:Complex Modulus"
] | [] |
proofwiki-1764 | Complex Modulus of Product of Complex Numbers | Let $z_1, z_2 \in \C$ be complex numbers.
Let $\cmod z$ be the modulus of $z$.
Then:
:$\cmod {z_1 z_2} = \cmod {z_1} \cdot \cmod {z_2}$ | Let $\overline z$ denote the complex conjugate of $z$.
Then:
{{begin-eqn}}
{{eqn | l = \cmod {z_1 z_2}
| r = \sqrt {\paren {z_1 z_2} \overline {\paren {z_1 z_2} } }
| c = Modulus in Terms of Conjugate
}}
{{eqn | r = \sqrt {z_1 \overline {z_1} z_2 \overline {z_2} }
| c = Product of Complex Conjugates,... | Let $z_1, z_2 \in \C$ be [[Definition:Complex Number|complex numbers]].
Let $\cmod z$ be the [[Definition:Complex Modulus|modulus]] of $z$.
Then:
:$\cmod {z_1 z_2} = \cmod {z_1} \cdot \cmod {z_2}$ | Let $\overline z$ denote the [[Definition:Complex Conjugate|complex conjugate]] of $z$.
Then:
{{begin-eqn}}
{{eqn | l = \cmod {z_1 z_2}
| r = \sqrt {\paren {z_1 z_2} \overline {\paren {z_1 z_2} } }
| c = [[Modulus in Terms of Conjugate]]
}}
{{eqn | r = \sqrt {z_1 \overline {z_1} z_2 \overline {z_2} }
... | Complex Modulus of Product of Complex Numbers/Proof 2 | https://proofwiki.org/wiki/Complex_Modulus_of_Product_of_Complex_Numbers | https://proofwiki.org/wiki/Complex_Modulus_of_Product_of_Complex_Numbers/Proof_2 | [
"Complex Modulus",
"Complex Multiplication",
"Complex Modulus of Product of Complex Numbers"
] | [
"Definition:Complex Number",
"Definition:Complex Modulus"
] | [
"Definition:Complex Conjugate",
"Modulus in Terms of Conjugate",
"Product of Complex Conjugates",
"Complex Multiplication is Commutative",
"Exponent Combination Laws/Power of Product"
] |
proofwiki-1765 | Complex Modulus of Product of Complex Numbers | Let $z_1, z_2 \in \C$ be complex numbers.
Let $\cmod z$ be the modulus of $z$.
Then:
:$\cmod {z_1 z_2} = \cmod {z_1} \cdot \cmod {z_2}$ | Let:
:$z_1 = r_1 \paren {\cos \theta_1 + i \sin \theta_1}$
:$z_2 = r_2 \paren {\cos \theta_2 + i \sin \theta_2}$
Then:
{{begin-eqn}}
{{eqn | l = \cmod {z_1 z_2}
| r = \cmod {r_1 \paren {\cos \theta_1 + i \sin \theta_1} r_2 \paren {\cos \theta_2 + i \sin \theta_2} }
| c = {{Defof|Polar Form of Complex Number... | Let $z_1, z_2 \in \C$ be [[Definition:Complex Number|complex numbers]].
Let $\cmod z$ be the [[Definition:Complex Modulus|modulus]] of $z$.
Then:
:$\cmod {z_1 z_2} = \cmod {z_1} \cdot \cmod {z_2}$ | Let:
:$z_1 = r_1 \paren {\cos \theta_1 + i \sin \theta_1}$
:$z_2 = r_2 \paren {\cos \theta_2 + i \sin \theta_2}$
Then:
{{begin-eqn}}
{{eqn | l = \cmod {z_1 z_2}
| r = \cmod {r_1 \paren {\cos \theta_1 + i \sin \theta_1} r_2 \paren {\cos \theta_2 + i \sin \theta_2} }
| c = {{Defof|Polar Form of Complex Numbe... | Complex Modulus of Product of Complex Numbers/Proof 3 | https://proofwiki.org/wiki/Complex_Modulus_of_Product_of_Complex_Numbers | https://proofwiki.org/wiki/Complex_Modulus_of_Product_of_Complex_Numbers/Proof_3 | [
"Complex Modulus",
"Complex Multiplication",
"Complex Modulus of Product of Complex Numbers"
] | [
"Definition:Complex Number",
"Definition:Complex Modulus"
] | [
"Product of Complex Numbers in Polar Form"
] |
proofwiki-1766 | Sum of Complex Conjugates | Let $z_1, z_2 \in \C$ be complex numbers.
Let $\overline z$ denote the complex conjugate of the complex number $z$.
Then:
:$\overline {z_1 + z_2} = \overline {z_1} + \overline {z_2}$ | Let $z_1 = x_1 + i y_1, z_2 = x_2 + i y_2$.
Then:
{{begin-eqn}}
{{eqn | l = \overline {z_1 + z_2}
| r = \overline {\paren {x_1 + x_2} + i \paren {y_1 + y_2} }
| c =
}}
{{eqn | r = \paren {x_1 + x_2} - i \paren {y_1 + y_2}
| c = {{Defof|Complex Conjugate}}
}}
{{eqn | r = \paren {x_1 - i y_1} + \paren ... | Let $z_1, z_2 \in \C$ be [[Definition:Complex Number|complex numbers]].
Let $\overline z$ denote the [[Definition:Complex Conjugate|complex conjugate]] of the [[Definition:Complex Number|complex number]] $z$.
Then:
:$\overline {z_1 + z_2} = \overline {z_1} + \overline {z_2}$ | Let $z_1 = x_1 + i y_1, z_2 = x_2 + i y_2$.
Then:
{{begin-eqn}}
{{eqn | l = \overline {z_1 + z_2}
| r = \overline {\paren {x_1 + x_2} + i \paren {y_1 + y_2} }
| c =
}}
{{eqn | r = \paren {x_1 + x_2} - i \paren {y_1 + y_2}
| c = {{Defof|Complex Conjugate}}
}}
{{eqn | r = \paren {x_1 - i y_1} + \pare... | Sum of Complex Conjugates | https://proofwiki.org/wiki/Sum_of_Complex_Conjugates | https://proofwiki.org/wiki/Sum_of_Complex_Conjugates | [
"Complex Conjugates",
"Complex Addition"
] | [
"Definition:Complex Number",
"Definition:Complex Conjugate",
"Definition:Complex Number"
] | [] |
proofwiki-1767 | Product of Complex Conjugates | Let $z_1, z_2 \in \C$ be complex numbers.
Let $\overline z$ denote the complex conjugate of the complex number $z$.
Then:
:$\overline {z_1 z_2} = \overline {z_1} \cdot \overline {z_2}$ | Let $z_1 = x_1 + i y_1$ and $z_2 = x_2 + i y_2$, where $x_1, y_1, x_2, y_2 \in \R$.
Then:
{{begin-eqn}}
{{eqn | l = \overline {z_1 z_2}
| r = \overline {\paren {x_1 x_2 - y_1 y_2} + i \paren {x_2 y_1 + x_1 y_2} }
| c = {{Defof|Complex Multiplication}}
}}
{{eqn | r = \paren {x_1 x_2 - y_1 y_2} - i \paren {x_... | Let $z_1, z_2 \in \C$ be [[Definition:Complex Number|complex numbers]].
Let $\overline z$ denote the [[Definition:Complex Conjugate|complex conjugate]] of the [[Definition:Complex Number|complex number]] $z$.
Then:
:$\overline {z_1 z_2} = \overline {z_1} \cdot \overline {z_2}$ | Let $z_1 = x_1 + i y_1$ and $z_2 = x_2 + i y_2$, where $x_1, y_1, x_2, y_2 \in \R$.
Then:
{{begin-eqn}}
{{eqn | l = \overline {z_1 z_2}
| r = \overline {\paren {x_1 x_2 - y_1 y_2} + i \paren {x_2 y_1 + x_1 y_2} }
| c = {{Defof|Complex Multiplication}}
}}
{{eqn | r = \paren {x_1 x_2 - y_1 y_2} - i \paren {... | Product of Complex Conjugates | https://proofwiki.org/wiki/Product_of_Complex_Conjugates | https://proofwiki.org/wiki/Product_of_Complex_Conjugates | [
"Product of Complex Conjugates",
"Complex Conjugates",
"Complex Multiplication"
] | [
"Definition:Complex Number",
"Definition:Complex Conjugate",
"Definition:Complex Number"
] | [] |
proofwiki-1768 | Product of Complex Conjugates | Let $z_1, z_2 \in \C$ be complex numbers.
Let $\overline z$ denote the complex conjugate of the complex number $z$.
Then:
:$\overline {z_1 z_2} = \overline {z_1} \cdot \overline {z_2}$ | From Product of Complex Conjugates: General Result:
:$\ds \overline {\prod_{j \mathop = 1}^n z_j} = \prod_{j \mathop = 1}^n \overline {z_j}$
The result follows by setting $n = 3$. | Let $z_1, z_2 \in \C$ be [[Definition:Complex Number|complex numbers]].
Let $\overline z$ denote the [[Definition:Complex Conjugate|complex conjugate]] of the [[Definition:Complex Number|complex number]] $z$.
Then:
:$\overline {z_1 z_2} = \overline {z_1} \cdot \overline {z_2}$ | From [[Product of Complex Conjugates/General Result|Product of Complex Conjugates: General Result]]:
:$\ds \overline {\prod_{j \mathop = 1}^n z_j} = \prod_{j \mathop = 1}^n \overline {z_j}$
The result follows by setting $n = 3$. | Product of Complex Conjugates/Examples/3 Arguments/Proof 2 | https://proofwiki.org/wiki/Product_of_Complex_Conjugates | https://proofwiki.org/wiki/Product_of_Complex_Conjugates/Examples/3_Arguments/Proof_2 | [
"Product of Complex Conjugates",
"Complex Conjugates",
"Complex Multiplication"
] | [
"Definition:Complex Number",
"Definition:Complex Conjugate",
"Definition:Complex Number"
] | [
"Product of Complex Conjugates/General Result"
] |
proofwiki-1769 | Sum of Complex Number with Conjugate | Let $z \in \C$ be a complex number.
Let $\overline z$ be the complex conjugate of $z$.
Let $\map \Re z$ be the real part of $z$.
Then:
:$z + \overline z = 2 \, \map \Re z$ | Let $z = x + i y$.
Then:
{{begin-eqn}}
{{eqn | l = z + \overline z
| r = \paren {x + i y} + \paren {x - i y}
| c = {{Defof|Complex Conjugate}}
}}
{{eqn | r = 2 x
}}
{{eqn | r = 2 \, \map \Re z
| c = {{Defof|Real Part}}
}}
{{end-eqn}}
{{qed}} | Let $z \in \C$ be a [[Definition:Complex Number|complex number]].
Let $\overline z$ be the [[Definition:Complex Conjugate|complex conjugate]] of $z$.
Let $\map \Re z$ be the [[Definition:Real Part|real part]] of $z$.
Then:
:$z + \overline z = 2 \, \map \Re z$ | Let $z = x + i y$.
Then:
{{begin-eqn}}
{{eqn | l = z + \overline z
| r = \paren {x + i y} + \paren {x - i y}
| c = {{Defof|Complex Conjugate}}
}}
{{eqn | r = 2 x
}}
{{eqn | r = 2 \, \map \Re z
| c = {{Defof|Real Part}}
}}
{{end-eqn}}
{{qed}} | Sum of Complex Number with Conjugate | https://proofwiki.org/wiki/Sum_of_Complex_Number_with_Conjugate | https://proofwiki.org/wiki/Sum_of_Complex_Number_with_Conjugate | [
"Complex Conjugates"
] | [
"Definition:Complex Number",
"Definition:Complex Conjugate",
"Definition:Complex Number/Real Part"
] | [] |
proofwiki-1770 | Difference of Complex Number with Conjugate | Let $z \in \C$ be a complex number.
Let $\overline z$ be the complex conjugate of $z$.
Let $\map \Im z$ be the imaginary part of $z$.
Then
:$z - \overline z = 2 i \, \map \Im z$ | Let $z = x + i y$.
Then:
{{begin-eqn}}
{{eqn | l = z - \overline z
| r = \paren {x + i y} - \paren {x - i y}
| c = {{Defof|Complex Conjugate}}
}}
{{eqn | r = x + i y - x + i y
}}
{{eqn | r = 2 i y
}}
{{eqn | r = 2 i \, \map \Im z
| c = {{Defof|Imaginary Part}}
}}
{{end-eqn}}
{{qed}} | Let $z \in \C$ be a [[Definition:Complex Number|complex number]].
Let $\overline z$ be the [[Definition:Complex Conjugate|complex conjugate]] of $z$.
Let $\map \Im z$ be the [[Definition:Imaginary Part|imaginary part]] of $z$.
Then
:$z - \overline z = 2 i \, \map \Im z$ | Let $z = x + i y$.
Then:
{{begin-eqn}}
{{eqn | l = z - \overline z
| r = \paren {x + i y} - \paren {x - i y}
| c = {{Defof|Complex Conjugate}}
}}
{{eqn | r = x + i y - x + i y
}}
{{eqn | r = 2 i y
}}
{{eqn | r = 2 i \, \map \Im z
| c = {{Defof|Imaginary Part}}
}}
{{end-eqn}}
{{qed}} | Difference of Complex Number with Conjugate | https://proofwiki.org/wiki/Difference_of_Complex_Number_with_Conjugate | https://proofwiki.org/wiki/Difference_of_Complex_Number_with_Conjugate | [
"Complex Conjugates"
] | [
"Definition:Complex Number",
"Definition:Complex Conjugate",
"Definition:Complex Number/Imaginary Part"
] | [] |
proofwiki-1771 | Complex Number equals Conjugate iff Wholly Real | Let $z \in \C$ be a complex number.
Let $\overline z$ be the complex conjugate of $z$.
Then $z = \overline z$ {{iff}} $z$ is wholly real. | Let $z = x + i y$.
Then:
{{begin-eqn}}
{{eqn | l = z
| r = \overline z
| c =
}}
{{eqn | ll= \leadsto
| l = x + i y
| r = x - i y
| c = {{Defof|Complex Conjugate}}
}}
{{eqn | ll= \leadsto
| l = +y
| r = -y
| c =
}}
{{eqn | ll= \leadsto
| l = y
| r = 0
|... | Let $z \in \C$ be a [[Definition:Complex Number|complex number]].
Let $\overline z$ be the [[Definition:Complex Conjugate|complex conjugate]] of $z$.
Then $z = \overline z$ {{iff}} $z$ is [[Definition:Wholly Real|wholly real]]. | Let $z = x + i y$.
Then:
{{begin-eqn}}
{{eqn | l = z
| r = \overline z
| c =
}}
{{eqn | ll= \leadsto
| l = x + i y
| r = x - i y
| c = {{Defof|Complex Conjugate}}
}}
{{eqn | ll= \leadsto
| l = +y
| r = -y
| c =
}}
{{eqn | ll= \leadsto
| l = y
| r = 0
... | Complex Number equals Conjugate iff Wholly Real | https://proofwiki.org/wiki/Complex_Number_equals_Conjugate_iff_Wholly_Real | https://proofwiki.org/wiki/Complex_Number_equals_Conjugate_iff_Wholly_Real | [
"Complex Conjugates"
] | [
"Definition:Complex Number",
"Definition:Complex Conjugate",
"Definition:Complex Number/Wholly Real"
] | [
"Definition:Complex Number/Wholly Real",
"Definition:Complex Number/Wholly Real"
] |
proofwiki-1772 | Complex Numbers cannot be Ordered Compatibly with Ring Structure | Let $\struct {\C, +, \times}$ be the field of complex numbers.
There exists no total ordering on $\struct {\C, +, \times}$ which is compatible with the structure of $\struct {\C, +, \times}$. | {{AimForCont}} there exists a relation $\preceq$ on $\C$ which is ordering compatible with the ring structure of $\C$.
That is:
:$(1): \quad z \ne 0 \implies 0 \prec z \lor z \prec 0$, but not both
:$(2): \quad 0 \prec z_1, z_2 \implies 0 \prec z_1 z_2 \land 0 \prec z_1 + z_2$
By Totally Ordered Ring Zero Precedes Elem... | Let $\struct {\C, +, \times}$ be the [[Definition:Field of Complex Numbers|field of complex numbers]].
There exists no [[Definition:Total Ordering|total ordering]] on $\struct {\C, +, \times}$ which is [[Definition:Ordering Compatible with Ring Structure|compatible with the structure]] of $\struct {\C, +, \times}$. | {{AimForCont}} there exists a [[Definition:Endorelation|relation]] $\preceq$ on $\C$ which is [[Definition:Ordering Compatible with Ring Structure|ordering compatible with the ring structure of $\C$]].
That is:
:$(1): \quad z \ne 0 \implies 0 \prec z \lor z \prec 0$, but not both
:$(2): \quad 0 \prec z_1, z_2 \implie... | Complex Numbers cannot be Ordered Compatibly with Ring Structure/Proof 1 | https://proofwiki.org/wiki/Complex_Numbers_cannot_be_Ordered_Compatibly_with_Ring_Structure | https://proofwiki.org/wiki/Complex_Numbers_cannot_be_Ordered_Compatibly_with_Ring_Structure/Proof_1 | [
"Complex Numbers",
"Complex Numbers cannot be Ordered Compatibly with Ring Structure"
] | [
"Definition:Field of Complex Numbers",
"Definition:Total Ordering",
"Definition:Ordering Compatible with Ring Structure"
] | [
"Definition:Endorelation",
"Definition:Ordering Compatible with Ring Structure",
"Totally Ordered Ring Zero Precedes Element or its Inverse",
"Proof by Cases",
"Definition:Contradiction",
"Proof by Contradiction"
] |
proofwiki-1773 | Complex Numbers cannot be Ordered Compatibly with Ring Structure | Let $\struct {\C, +, \times}$ be the field of complex numbers.
There exists no total ordering on $\struct {\C, +, \times}$ which is compatible with the structure of $\struct {\C, +, \times}$. | {{AimForCont}} such a total ordering $\preceq$ exists.
By the definition of a total ordering, $\preceq$ is connected.
That is:
:$0 \preceq i \lor i \preceq 0$
Using Proof by Cases, we will prove that:
:$0 \preceq -1$
;Case 1
Assume that $0 \preceq i$.
By definition of an ordering compatible with the ring structure of $... | Let $\struct {\C, +, \times}$ be the [[Definition:Field of Complex Numbers|field of complex numbers]].
There exists no [[Definition:Total Ordering|total ordering]] on $\struct {\C, +, \times}$ which is [[Definition:Ordering Compatible with Ring Structure|compatible with the structure]] of $\struct {\C, +, \times}$. | {{AimForCont}} such a [[Definition:Total Ordering|total ordering]] $\preceq$ exists.
By the definition of a [[Definition:Total Ordering/Definition 1|total ordering]], $\preceq$ is [[Definition:Connected Relation|connected]].
That is:
:$0 \preceq i \lor i \preceq 0$
Using [[Proof by Cases]], we will prove that:
:$0 ... | Complex Numbers cannot be Ordered Compatibly with Ring Structure/Proof 2 | https://proofwiki.org/wiki/Complex_Numbers_cannot_be_Ordered_Compatibly_with_Ring_Structure | https://proofwiki.org/wiki/Complex_Numbers_cannot_be_Ordered_Compatibly_with_Ring_Structure/Proof_2 | [
"Complex Numbers",
"Complex Numbers cannot be Ordered Compatibly with Ring Structure"
] | [
"Definition:Field of Complex Numbers",
"Definition:Total Ordering",
"Definition:Ordering Compatible with Ring Structure"
] | [
"Definition:Total Ordering",
"Definition:Total Ordering/Definition 1",
"Definition:Connected Relation",
"Proof by Cases",
"Definition:Ordering Compatible with Ring Structure",
"Definition:Relation Compatible with Operation",
"Definition:Ordering Compatible with Ring Structure",
"Proof by Cases",
"De... |
proofwiki-1774 | Complex Numbers cannot be Ordered Compatibly with Ring Structure | Let $\struct {\C, +, \times}$ be the field of complex numbers.
There exists no total ordering on $\struct {\C, +, \times}$ which is compatible with the structure of $\struct {\C, +, \times}$. | From Complex Numbers form Integral Domain, $\struct {\C, +, \times}$ is an integral domain.
{{AimForCont}} that $\struct {\C, +, \times}$ can be ordered.
Thus, by definition, it possesses a (strict) positivity property $P$.
Then from Strict Positivity Property induces Total Ordering, let $\le$ be the total ordering ind... | Let $\struct {\C, +, \times}$ be the [[Definition:Field of Complex Numbers|field of complex numbers]].
There exists no [[Definition:Total Ordering|total ordering]] on $\struct {\C, +, \times}$ which is [[Definition:Ordering Compatible with Ring Structure|compatible with the structure]] of $\struct {\C, +, \times}$. | From [[Complex Numbers form Integral Domain]], $\struct {\C, +, \times}$ is an [[Definition:Integral Domain|integral domain]].
{{AimForCont}} that $\struct {\C, +, \times}$ can be [[Definition:Ordered Integral Domain|ordered]].
Thus, by definition, it possesses a [[Definition:Strict Positivity Property|(strict) posi... | Complex Numbers cannot be Ordered Compatibly with Ring Structure/Proof 3 | https://proofwiki.org/wiki/Complex_Numbers_cannot_be_Ordered_Compatibly_with_Ring_Structure | https://proofwiki.org/wiki/Complex_Numbers_cannot_be_Ordered_Compatibly_with_Ring_Structure/Proof_3 | [
"Complex Numbers",
"Complex Numbers cannot be Ordered Compatibly with Ring Structure"
] | [
"Definition:Field of Complex Numbers",
"Definition:Total Ordering",
"Definition:Ordering Compatible with Ring Structure"
] | [
"Complex Numbers form Integral Domain",
"Definition:Integral Domain",
"Definition:Ordered Integral Domain",
"Definition:Strict Positivity Property",
"Strict Positivity Property induces Total Ordering",
"Definition:Total Ordering Induced by Strict Positivity Property",
"Unity of Ordered Integral Domain i... |
proofwiki-1775 | Limit of Complex Function is Unique | Let $f: S \to \C$ be a complex function.
Let $z_0$ be a limit point of $S$.
Suppose that $\ds \lim_{z \mathop \to z_0} \map f z = L$.
Then that limit $L$ is unique. | {{AimForCont}} $L' \ne L$ is another limit point of $\map f z$ at $z_0$.
Let us take $\epsilon = \dfrac {\cmod {L - L'} } 2$.
Then we can find $\delta_1 > 0, \delta_2 > 0$ such that:
:$z \in S, 0 < \cmod {z - z_0} < \delta_1 \implies \cmod {\map f z - L} < \epsilon$
:$z \in S, 0 < \cmod {z - z_0} < \delta_2 \implies \c... | Let $f: S \to \C$ be a [[Definition:Complex Function|complex function]].
Let $z_0$ be a [[Definition:Limit Point (Complex Analysis)|limit point]] of $S$.
Suppose that $\ds \lim_{z \mathop \to z_0} \map f z = L$.
Then that limit $L$ is [[Definition:Unique|unique]]. | {{AimForCont}} $L' \ne L$ is another [[Definition:Limit Point (Complex Analysis)|limit point]] of $\map f z$ at $z_0$.
Let us take $\epsilon = \dfrac {\cmod {L - L'} } 2$.
Then we can find $\delta_1 > 0, \delta_2 > 0$ such that:
:$z \in S, 0 < \cmod {z - z_0} < \delta_1 \implies \cmod {\map f z - L} < \epsilon$
:$z ... | Limit of Complex Function is Unique | https://proofwiki.org/wiki/Limit_of_Complex_Function_is_Unique | https://proofwiki.org/wiki/Limit_of_Complex_Function_is_Unique | [
"Limits of Complex Functions"
] | [
"Definition:Complex Function",
"Definition:Limit Point/Complex Analysis",
"Definition:Unique"
] | [
"Definition:Limit Point/Complex Analysis",
"Definition:Limit Point/Complex Analysis",
"Triangle Inequality",
"Definition:Contradiction",
"Proof by Contradiction",
"Category:Limits of Complex Functions"
] |
proofwiki-1776 | Complex Plane is Metric Space | Let $\C$ be the set of all complex numbers.
Let $d: \C \times \C \to \R$ be the function defined as:
:$\map d {z_1, z_2} = \size {z_1 - z_2}$
where $\size z$ is the modulus of $z$.
Then $d$ is a metric on $\C$ and so $\struct {\C, d}$ is a metric space. | Let $z_1 = x_1 + i y_1, z_2 = x_2 + i y_2$.
From the definition of modulus:
:$\size {z_1 - z_2} = \sqrt {\paren {x_1 - x_2}^2 + \paren {y_1 - y_2}^2}$
This is the euclidean metric on the real number plane.
This is shown in Euclidean Metric on Real Vector Space is Metric to be a metric.
Thus the complex plane is a 2-dim... | Let $\C$ be the [[Definition:Set|set]] of all [[Definition:Complex Number|complex numbers]].
Let $d: \C \times \C \to \R$ be the [[Definition:Mapping|function]] defined as:
:$\map d {z_1, z_2} = \size {z_1 - z_2}$
where $\size z$ is the [[Definition:Complex Modulus|modulus]] of $z$.
Then $d$ is a [[Definition:Metric|... | Let $z_1 = x_1 + i y_1, z_2 = x_2 + i y_2$.
From the definition of [[Definition:Complex Modulus|modulus]]:
:$\size {z_1 - z_2} = \sqrt {\paren {x_1 - x_2}^2 + \paren {y_1 - y_2}^2}$
This is the [[Definition:Euclidean Metric on Real Number Plane|euclidean metric]] on the [[Definition:Real Number Plane|real number pla... | Complex Plane is Metric Space | https://proofwiki.org/wiki/Complex_Plane_is_Metric_Space | https://proofwiki.org/wiki/Complex_Plane_is_Metric_Space | [
"Complex Plane",
"Examples of Metric Spaces"
] | [
"Definition:Set",
"Definition:Complex Number",
"Definition:Mapping",
"Definition:Complex Modulus",
"Definition:Metric Space/Metric",
"Definition:Metric Space"
] | [
"Definition:Complex Modulus",
"Definition:Euclidean Metric/Real Number Plane",
"Definition:Real Number Plane",
"Euclidean Metric on Real Vector Space is Metric",
"Definition:Metric Space/Metric",
"Definition:Complex Number/Complex Plane",
"Definition:Dimension (Geometry)",
"Definition:Euclidean Space"... |
proofwiki-1777 | Real Number Line is Metric Space | Let $\R$ be the real number line.
Let $d: \R \times \R \to \R$ be defined as:
:$\map d {x_1, x_2} = \size {x_1 - x_2}$
where $\size x$ is the absolute value of $x$.
Then $d$ is a metric on $\R$ and so $\struct {\R, d}$ is a metric space. | === Proof of {{Metric-space-axiom|1|nolink}} ===
{{begin-eqn}}
{{eqn | l = \map d {x, x}
| r = \size {x - x}
| c = Definition of $d$
}}
{{eqn | r = 0
| c = {{Defof|Absolute Value}}
}}
{{end-eqn}}
So {{Metric-space-axiom|1}} holds for $d$.
{{qed|lemma}} | Let $\R$ be the [[Definition:Real Number Line|real number line]].
Let $d: \R \times \R \to \R$ be defined as:
:$\map d {x_1, x_2} = \size {x_1 - x_2}$
where $\size x$ is the [[Definition:Absolute Value|absolute value]] of $x$.
Then $d$ is a [[Definition:Metric|metric]] on $\R$ and so $\struct {\R, d}$ is a [[Definit... | === Proof of {{Metric-space-axiom|1|nolink}} ===
{{begin-eqn}}
{{eqn | l = \map d {x, x}
| r = \size {x - x}
| c = Definition of $d$
}}
{{eqn | r = 0
| c = {{Defof|Absolute Value}}
}}
{{end-eqn}}
So {{Metric-space-axiom|1}} holds for $d$.
{{qed|lemma}} | Real Number Line is Metric Space | https://proofwiki.org/wiki/Real_Number_Line_is_Metric_Space | https://proofwiki.org/wiki/Real_Number_Line_is_Metric_Space | [
"Real Number Line with Euclidean Metric",
"Real Number Line",
"Examples of Metric Spaces"
] | [
"Definition:Real Number/Real Number Line",
"Definition:Absolute Value",
"Definition:Metric Space/Metric",
"Definition:Metric Space"
] | [] |
proofwiki-1778 | P-Product Metric on Real Vector Space is Metric | Let $\R^n$ be an $n$-dimensional real vector space.
Let $p \in \R_{\ge 1}$.
Let $d_p: \R^n \times \R^n \to \R$ be the $p$-product metric on $\R^n$:
:$\ds \map {d_p} {x, y} := \paren {\sum_{i \mathop = 1}^n \size {x_i - y_i}^p}^{\frac 1 p}$
where $x = \tuple {x_1, x_2, \ldots, x_n}, y = \tuple {y_1, y_2, \ldots, y_n} \i... | === Proof of {{Metric-space-axiom|1|nolink}} ===
{{begin-eqn}}
{{eqn | l = \map {d_p} {x, x}
| r = \paren {\sum_{i \mathop = 1}^n \size {x_i - x_i}^p}^{\frac 1 p}
| c = Definition of $d_p$
}}
{{eqn | r = \paren {\sum_{i \mathop = 1}^n 0^p}^{\frac 1 p}
| c =
}}
{{eqn | r = 0
| c =
}}
{{end-eqn}... | Let $\R^n$ be an [[Definition:Dimension of Vector Space|$n$-dimensional]] [[Definition:Real Vector Space|real vector space]].
Let $p \in \R_{\ge 1}$.
Let $d_p: \R^n \times \R^n \to \R$ be the [[Definition:P-Product Metric on Real Vector Space|$p$-product metric]] on $\R^n$:
:$\ds \map {d_p} {x, y} := \paren {\sum_{... | === Proof of {{Metric-space-axiom|1|nolink}} ===
{{begin-eqn}}
{{eqn | l = \map {d_p} {x, x}
| r = \paren {\sum_{i \mathop = 1}^n \size {x_i - x_i}^p}^{\frac 1 p}
| c = Definition of $d_p$
}}
{{eqn | r = \paren {\sum_{i \mathop = 1}^n 0^p}^{\frac 1 p}
| c =
}}
{{eqn | r = 0
| c =
}}
{{end-eqn... | P-Product Metric on Real Vector Space is Metric/Proof 1 | https://proofwiki.org/wiki/P-Product_Metric_on_Real_Vector_Space_is_Metric | https://proofwiki.org/wiki/P-Product_Metric_on_Real_Vector_Space_is_Metric/Proof_1 | [
"P-Product Metrics",
"P-Product Metric on Real Vector Space is Metric"
] | [
"Definition:Dimension of Vector Space",
"Definition:Real Vector Space",
"Definition:P-Product Metric/Real Vector Space",
"Definition:Metric Space/Metric"
] | [
"Minkowski's Inequality for Sums",
"Triangle Inequality/Real Numbers"
] |
proofwiki-1779 | P-Product Metric on Real Vector Space is Metric | Let $\R^n$ be an $n$-dimensional real vector space.
Let $p \in \R_{\ge 1}$.
Let $d_p: \R^n \times \R^n \to \R$ be the $p$-product metric on $\R^n$:
:$\ds \map {d_p} {x, y} := \paren {\sum_{i \mathop = 1}^n \size {x_i - y_i}^p}^{\frac 1 p}$
where $x = \tuple {x_1, x_2, \ldots, x_n}, y = \tuple {y_1, y_2, \ldots, y_n} \i... | This is an instance of $p$-Product Metric is Metric.
{{qed}} | Let $\R^n$ be an [[Definition:Dimension of Vector Space|$n$-dimensional]] [[Definition:Real Vector Space|real vector space]].
Let $p \in \R_{\ge 1}$.
Let $d_p: \R^n \times \R^n \to \R$ be the [[Definition:P-Product Metric on Real Vector Space|$p$-product metric]] on $\R^n$:
:$\ds \map {d_p} {x, y} := \paren {\sum_{... | This is an instance of [[P-Product Metric is Metric|$p$-Product Metric is Metric]].
{{qed}} | P-Product Metric on Real Vector Space is Metric/Proof 2 | https://proofwiki.org/wiki/P-Product_Metric_on_Real_Vector_Space_is_Metric | https://proofwiki.org/wiki/P-Product_Metric_on_Real_Vector_Space_is_Metric/Proof_2 | [
"P-Product Metrics",
"P-Product Metric on Real Vector Space is Metric"
] | [
"Definition:Dimension of Vector Space",
"Definition:Real Vector Space",
"Definition:P-Product Metric/Real Vector Space",
"Definition:Metric Space/Metric"
] | [
"P-Product Metric is Metric"
] |
proofwiki-1780 | Taxicab Metric is Metric | The taxicab metric is a metric. | From the definition, the taxicab metric is as follows:
Let $M_{1'} = \struct {A_{1'}, d_{1'} }, M_{2'} = \struct {A_{2'}, d_{2'} }, \ldots, M_{n'} = \struct {A_{n'}, d_{n'} }$ be a finite number of metric spaces.
Let $\AA$ be the Cartesian product $\ds \prod_{i \mathop = 1}^n A_{i'}$.
The taxicab metric on $\AA$ is:
:$... | The [[Definition:Taxicab Metric|taxicab metric]] is a [[Definition:Metric|metric]]. | From the definition, the [[Definition:Taxicab Metric|taxicab metric]] is as follows:
Let $M_{1'} = \struct {A_{1'}, d_{1'} }, M_{2'} = \struct {A_{2'}, d_{2'} }, \ldots, M_{n'} = \struct {A_{n'}, d_{n'} }$ be a [[Definition:Finite Set|finite number]] of [[Definition:Metric Space|metric spaces]].
Let $\AA$ be the [[De... | Taxicab Metric is Metric/Proof 1 | https://proofwiki.org/wiki/Taxicab_Metric_is_Metric | https://proofwiki.org/wiki/Taxicab_Metric_is_Metric/Proof_1 | [
"Taxicab Metric",
"Taxicab Metric is Metric"
] | [
"Definition:Taxicab Metric",
"Definition:Metric Space/Metric"
] | [
"Definition:Taxicab Metric",
"Definition:Finite Set",
"Definition:Metric Space",
"Definition:Cartesian Product/Finite",
"Definition:Taxicab Metric"
] |
proofwiki-1781 | Taxicab Metric is Metric | The taxicab metric is a metric. | Follows directly from P-Product Metric is Metric, where in this case $p = 1$.
{{qed}} | The [[Definition:Taxicab Metric|taxicab metric]] is a [[Definition:Metric|metric]]. | Follows directly from [[P-Product Metric is Metric]], where in this case $p = 1$.
{{qed}} | Taxicab Metric is Metric/Proof 2 | https://proofwiki.org/wiki/Taxicab_Metric_is_Metric | https://proofwiki.org/wiki/Taxicab_Metric_is_Metric/Proof_2 | [
"Taxicab Metric",
"Taxicab Metric is Metric"
] | [
"Definition:Taxicab Metric",
"Definition:Metric Space/Metric"
] | [
"P-Product Metric is Metric"
] |
proofwiki-1782 | Zero and One are the only Consecutive Perfect Squares | If $n$ is a perfect square other than $0$, then $n+1$ is not a perfect square. | Let $x$ and $h$ be integers such that:
:$x^2 + 1 = \paren {x - h}^2$
Then:
{{begin-eqn}}
{{eqn | l = x^2 + 1
| r = \paren {x - h}^2
}}
{{eqn | l = 1
| r = -2 x h + h^2
}}
{{eqn | l = 2 x h
| r = h^2 - 1
}}
{{eqn | l = 2 x h
| r = \paren {h - 1} \paren {h + 1}
}}
{{end-eqn}}
We have that Consecut... | If $n$ is a [[Definition:Square Number|perfect square]] other than $0$, then $n+1$ is not a perfect square. | Let $x$ and $h$ be [[Definition:Integer|integers]] such that:
:$x^2 + 1 = \paren {x - h}^2$
Then:
{{begin-eqn}}
{{eqn | l = x^2 + 1
| r = \paren {x - h}^2
}}
{{eqn | l = 1
| r = -2 x h + h^2
}}
{{eqn | l = 2 x h
| r = h^2 - 1
}}
{{eqn | l = 2 x h
| r = \paren {h - 1} \paren {h + 1}
}}
{{end-eqn... | Zero and One are the only Consecutive Perfect Squares/Proof 1 | https://proofwiki.org/wiki/Zero_and_One_are_the_only_Consecutive_Perfect_Squares | https://proofwiki.org/wiki/Zero_and_One_are_the_only_Consecutive_Perfect_Squares/Proof_1 | [
"Square Numbers",
"Zero and One are the only Consecutive Perfect Squares"
] | [
"Definition:Square Number"
] | [
"Definition:Integer",
"Consecutive Integers are Coprime",
"Fundamental Theorem of Arithmetic",
"Definition:Prime Decomposition",
"Definition:Prime Factor",
"Definition:Integer",
"Definition:Prime Factor",
"Definition:Contradiction",
"Definition:Square Number"
] |
proofwiki-1783 | Zero and One are the only Consecutive Perfect Squares | If $n$ is a perfect square other than $0$, then $n+1$ is not a perfect square. | Suppose that $k, l \in \Z$ are such that their squares are consecutive.
That is:
:$l^2 - k^2 = 1$
Then we can factor the {{LHS}} as:
:$l^2 - k^2 = \paren {l + k} \paren {l - k}$
By Invertible Integers under Multiplication, it follows that:
:$l + k = \pm 1 = l - k$
Therefore, it must be that:
:$\paren {l + k} - \paren {... | If $n$ is a [[Definition:Square Number|perfect square]] other than $0$, then $n+1$ is not a perfect square. | Suppose that $k, l \in \Z$ are such that their [[Definition:Square Function|squares]] are consecutive.
That is:
:$l^2 - k^2 = 1$
Then we can factor the {{LHS}} as:
:$l^2 - k^2 = \paren {l + k} \paren {l - k}$
By [[Invertible Integers under Multiplication]], it follows that:
:$l + k = \pm 1 = l - k$
Therefore, i... | Zero and One are the only Consecutive Perfect Squares/Proof 2 | https://proofwiki.org/wiki/Zero_and_One_are_the_only_Consecutive_Perfect_Squares | https://proofwiki.org/wiki/Zero_and_One_are_the_only_Consecutive_Perfect_Squares/Proof_2 | [
"Square Numbers",
"Zero and One are the only Consecutive Perfect Squares"
] | [
"Definition:Square Number"
] | [
"Definition:Square/Function",
"Invertible Integers under Multiplication",
"Definition:Square Number"
] |
proofwiki-1784 | ProofWiki:Jokes | If I cannot open these cans of food, I will die. | Suppose not.
{{qed}} | If I cannot open these cans of food, I will die. | [[Proof by Contradiction|Suppose not]].
{{qed}} | ProofWiki:Jokes | https://proofwiki.org/wiki/ProofWiki:Jokes | https://proofwiki.org/wiki/ProofWiki:Jokes | [
"Jokes"
] | [] | [
"Proof by Contradiction"
] |
proofwiki-1785 | Finite Union of Bounded Subsets | Let $M = \struct {A, d}$ be a metric space.
Then the union of any finite number of bounded subsets of $M$ is itself bounded. | It is sufficient to prove this for two subsets, as the general result follows by induction.
Suppose $S_1$ and $S_2$ are bounded subsets of $M = \struct {A, d}$.
Let $a_1, a_2 \in A$.
Let $K_1, K_2 \in \R$ such that:
:$(1): \quad \forall x \in S_1: \map d {x, a_1} \le K_1$
:$(2): \quad \forall x \in S_2: \map d {x, a_2}... | Let $M = \struct {A, d}$ be a [[Definition:Metric Space|metric space]].
Then the [[Definition:Set Union|union]] of any [[Definition:Finite Set|finite]] number of [[Definition:Bounded Metric Space|bounded subsets]] of $M$ is itself [[Definition:Bounded Metric Space|bounded]]. | It is sufficient to prove this for two subsets, as the general result follows by [[Principle of Mathematical Induction|induction]].
Suppose $S_1$ and $S_2$ are [[Definition:Bounded Metric Space|bounded subsets]] of $M = \struct {A, d}$.
Let $a_1, a_2 \in A$.
Let $K_1, K_2 \in \R$ such that:
:$(1): \quad \forall x \i... | Finite Union of Bounded Subsets | https://proofwiki.org/wiki/Finite_Union_of_Bounded_Subsets | https://proofwiki.org/wiki/Finite_Union_of_Bounded_Subsets | [
"Boundedness"
] | [
"Definition:Metric Space",
"Definition:Set Union",
"Definition:Finite Set",
"Definition:Bounded Metric Space",
"Definition:Bounded Metric Space"
] | [
"Principle of Mathematical Induction",
"Definition:Bounded Metric Space"
] |
proofwiki-1786 | P-Product Metric is Metric | Let $M_{1'} = \struct {A_{1'}, d_{1'} }, M_{2'} = \struct {A_{2'}, d_{2'} }, \ldots, M_{n'} = \struct {A_{n'}, d_{n'} }$ be metric spaces.
Let $\ds \AA = \prod_{i \mathop = 1}^n A_{i'}$ be the cartesian product of $A_{1'}, A_{2'}, \ldots, A_{n'}$.
Let $p \in \R_{\ge 1}$.
Let $d_p: \AA \times \AA \to \R$ be the $p$-prod... | === Proof of {{Metric-space-axiom|1|nolink}} ===
{{begin-eqn}}
{{eqn | l = \map {d_p} {x, x}
| r = \paren {\sum_{i \mathop = 1}^n \paren {\map {d_{i'} } {x_i, x_i} }^p}^{\frac 1 p}
| c = Definition of $d_p$
}}
{{eqn | r = \paren {\sum_{i \mathop = 1}^n 0^p}^{\frac 1 p}
| c = as $d_{i'}$ fulfills {{Met... | Let $M_{1'} = \struct {A_{1'}, d_{1'} }, M_{2'} = \struct {A_{2'}, d_{2'} }, \ldots, M_{n'} = \struct {A_{n'}, d_{n'} }$ be [[Definition:Metric Space|metric spaces]].
Let $\ds \AA = \prod_{i \mathop = 1}^n A_{i'}$ be the [[Definition:Finite Cartesian Product|cartesian product]] of $A_{1'}, A_{2'}, \ldots, A_{n'}$.
Le... | === Proof of {{Metric-space-axiom|1|nolink}} ===
{{begin-eqn}}
{{eqn | l = \map {d_p} {x, x}
| r = \paren {\sum_{i \mathop = 1}^n \paren {\map {d_{i'} } {x_i, x_i} }^p}^{\frac 1 p}
| c = Definition of $d_p$
}}
{{eqn | r = \paren {\sum_{i \mathop = 1}^n 0^p}^{\frac 1 p}
| c = as $d_{i'}$ fulfills {{Me... | P-Product Metric is Metric | https://proofwiki.org/wiki/P-Product_Metric_is_Metric | https://proofwiki.org/wiki/P-Product_Metric_is_Metric | [
"P-Product Metrics"
] | [
"Definition:Metric Space",
"Definition:Cartesian Product/Finite",
"Definition:P-Product Metric/General Definition",
"Definition:Metric Space/Metric"
] | [
"Minkowski's Inequality for Sums"
] |
proofwiki-1787 | Euler Triangle Formula | Let $d$ be the distance between the incenter and the circumcenter of a triangle.
Then:
:$d^2 = R \paren {R - 2 \rho}$
where:
:$R$ is the circumradius
:$\rho$ is the inradius. | {{WLOG}}, it will be demonstrated that $BP = IP$.
Let $CP$ be the bisector of $\angle ACB$.
:$\angle ACP$ and $\angle ICB$ both subtend $AP$.
Hence indirectly by the Inscribed Angle Theorem:
:$\angle ACP = \angle ICB$
From Angles on Equal Arcs are Equal:
:$\angle ACP = \angle ABP$
and so:
:$\angle ABP = \angle ICB$
By ... | Let $d$ be the [[Definition:Distance between Points|distance]] between the [[Definition:Incenter of Triangle|incenter]] and the [[Definition:Circumcenter of Triangle|circumcenter]] of a [[Definition:Triangle (Geometry)|triangle]].
Then:
:$d^2 = R \paren {R - 2 \rho}$
where:
:$R$ is the [[Definition:Circumradius of Tri... | {{WLOG}}, it will be demonstrated that $BP = IP$.
Let $CP$ be the [[Definition:Bisection|bisector]] of $\angle ACB$.
:$\angle ACP$ and $\angle ICB$ both subtend $AP$.
Hence indirectly by the [[Inscribed Angle Theorem]]:
:$\angle ACP = \angle ICB$
From [[Angles on Equal Arcs are Equal]]:
:$\angle ACP = \angle ABP$
an... | Euler Triangle Formula/Lemma 2/Proof 1 | https://proofwiki.org/wiki/Euler_Triangle_Formula | https://proofwiki.org/wiki/Euler_Triangle_Formula/Lemma_2/Proof_1 | [
"Euler Triangle Formula",
"Circumcenters",
"Incircles of Triangles",
"Triangles"
] | [
"Definition:Distance between Points",
"Definition:Incircle of Triangle/Incenter",
"Definition:Circumcircle of Triangle/Circumcenter",
"Definition:Triangle (Geometry)",
"Definition:Circumcircle of Triangle/Circumradius",
"Definition:Incircle of Triangle/Inradius"
] | [
"Definition:Bisection",
"Inscribed Angle Theorem",
"Angles on Equal Arcs are Equal",
"Inscribing Circle in Triangle",
"Definition:Bisection",
"Sum of Angles of Triangle equals Two Right Angles",
"Definition:Supplementary Angles",
"Triangle with Two Equal Angles is Isosceles"
] |
proofwiki-1788 | Euler Triangle Formula | Let $d$ be the distance between the incenter and the circumcenter of a triangle.
Then:
:$d^2 = R \paren {R - 2 \rho}$
where:
:$R$ is the circumradius
:$\rho$ is the inradius. | We have {{hypothesis}}:
:$AI$, $BI$ and $CI$ bisect their respective angles.
Let the half-angles be:
:$\alpha = \dfrac 1 2 \angle CAB$
:$\beta = \dfrac 1 2 \angle ABC$
:$\gamma = \dfrac 1 2 \angle BCA$
:300px
{{begin-eqn}}
{{eqn | l = AP
| r = BP
| c = Equal Angles in Equal Circles
}}
{{eqn | l = \angle BAP... | Let $d$ be the [[Definition:Distance between Points|distance]] between the [[Definition:Incenter of Triangle|incenter]] and the [[Definition:Circumcenter of Triangle|circumcenter]] of a [[Definition:Triangle (Geometry)|triangle]].
Then:
:$d^2 = R \paren {R - 2 \rho}$
where:
:$R$ is the [[Definition:Circumradius of Tri... | We have {{hypothesis}}:
:$AI$, $BI$ and $CI$ [[Definition:Bisector|bisect]] their respective [[Definition:Angle|angles]].
Let the half-[[Definition:Angle|angles]] be:
:$\alpha = \dfrac 1 2 \angle CAB$
:$\beta = \dfrac 1 2 \angle ABC$
:$\gamma = \dfrac 1 2 \angle BCA$
:[[File:EulerTriangleLemma-2a.png|300px]]
{{begin... | Euler Triangle Formula/Lemma 2/Proof 2 | https://proofwiki.org/wiki/Euler_Triangle_Formula | https://proofwiki.org/wiki/Euler_Triangle_Formula/Lemma_2/Proof_2 | [
"Euler Triangle Formula",
"Circumcenters",
"Incircles of Triangles",
"Triangles"
] | [
"Definition:Distance between Points",
"Definition:Incircle of Triangle/Incenter",
"Definition:Circumcircle of Triangle/Circumcenter",
"Definition:Triangle (Geometry)",
"Definition:Circumcircle of Triangle/Circumradius",
"Definition:Incircle of Triangle/Inradius"
] | [
"Definition:Bisection/Bisector",
"Definition:Angle",
"Definition:Angle",
"File:EulerTriangleLemma-2a.png",
"Equal Angles in Equal Circles",
"Angles on Equal Arcs are Equal",
"Angles on Equal Arcs are Equal",
"Sum of Angles of Triangle equals Two Right Angles",
"Triangle with Two Equal Angles is Isos... |
proofwiki-1789 | Euler Triangle Formula | Let $d$ be the distance between the incenter and the circumcenter of a triangle.
Then:
:$d^2 = R \paren {R - 2 \rho}$
where:
:$R$ is the circumradius
:$\rho$ is the inradius. | === Lemma 1 ===
{{:Euler Triangle Formula/Lemma 1}}{{qed|lemma}}
=== Lemma 2 ===
{{:Euler Triangle Formula/Lemma 2}}{{qed|lemma}}
By {{Lemma|Euler Triangle Formula|1}}:
:$GI \cdot IJ = IP \cdot CI$
substituting:
:$IP \cdot CI = \paren {R + d} \paren {R - d}$
By {{Lemma|Euler Triangle Formula|2}}:
:$IP = PB$
and so:
:$G... | Let $d$ be the [[Definition:Distance between Points|distance]] between the [[Definition:Incenter of Triangle|incenter]] and the [[Definition:Circumcenter of Triangle|circumcenter]] of a [[Definition:Triangle (Geometry)|triangle]].
Then:
:$d^2 = R \paren {R - 2 \rho}$
where:
:$R$ is the [[Definition:Circumradius of Tri... | === [[Euler Triangle Formula/Lemma 1|Lemma 1]] ===
{{:Euler Triangle Formula/Lemma 1}}{{qed|lemma}}
=== [[Euler Triangle Formula/Lemma 2|Lemma 2]] ===
{{:Euler Triangle Formula/Lemma 2}}{{qed|lemma}}
By {{Lemma|Euler Triangle Formula|1}}:
:$GI \cdot IJ = IP \cdot CI$
substituting:
:$IP \cdot CI = \paren {R + d} \p... | Euler Triangle Formula/Proof 1 | https://proofwiki.org/wiki/Euler_Triangle_Formula | https://proofwiki.org/wiki/Euler_Triangle_Formula/Proof_1 | [
"Euler Triangle Formula",
"Circumcenters",
"Incircles of Triangles",
"Triangles"
] | [
"Definition:Distance between Points",
"Definition:Incircle of Triangle/Incenter",
"Definition:Circumcircle of Triangle/Circumcenter",
"Definition:Triangle (Geometry)",
"Definition:Circumcircle of Triangle/Circumradius",
"Definition:Incircle of Triangle/Inradius"
] | [
"Euler Triangle Formula/Lemma 1",
"Euler Triangle Formula/Lemma 2",
"Law of Sines",
"Axiom:Euclid's Common Notion 4",
"Radius at Right Angle to Tangent",
"Definition:Right Angle",
"Definition:Sine/Definition from Triangle",
"Difference of Two Squares"
] |
proofwiki-1790 | Euler Triangle Formula | Let $d$ be the distance between the incenter and the circumcenter of a triangle.
Then:
:$d^2 = R \paren {R - 2 \rho}$
where:
:$R$ is the circumradius
:$\rho$ is the inradius. | === Lemma 1 ===
{{:Euler Triangle Formula/Lemma 1}}{{qed|lemma}}
=== Lemma 2 ===
{{:Euler Triangle Formula/Lemma 2}}{{qed|lemma}}
{{begin-eqn}}
{{eqn | ll= \leadsto
| l = IP \cdot CI
| r = \paren {R + d} \cdot \paren {R - d}
| c = {{Lemma|Euler Triangle Formula|1}}
}}
{{eqn | l = IP
| r = PB
... | Let $d$ be the [[Definition:Distance between Points|distance]] between the [[Definition:Incenter of Triangle|incenter]] and the [[Definition:Circumcenter of Triangle|circumcenter]] of a [[Definition:Triangle (Geometry)|triangle]].
Then:
:$d^2 = R \paren {R - 2 \rho}$
where:
:$R$ is the [[Definition:Circumradius of Tri... | === [[Euler Triangle Formula/Lemma 1|Lemma 1]] ===
{{:Euler Triangle Formula/Lemma 1}}{{qed|lemma}}
=== [[Euler Triangle Formula/Lemma 2|Lemma 2]] ===
{{:Euler Triangle Formula/Lemma 2}}{{qed|lemma}}
{{begin-eqn}}
{{eqn | ll= \leadsto
| l = IP \cdot CI
| r = \paren {R + d} \cdot \paren {R - d}
| c =... | Euler Triangle Formula/Proof 2 | https://proofwiki.org/wiki/Euler_Triangle_Formula | https://proofwiki.org/wiki/Euler_Triangle_Formula/Proof_2 | [
"Euler Triangle Formula",
"Circumcenters",
"Incircles of Triangles",
"Triangles"
] | [
"Definition:Distance between Points",
"Definition:Incircle of Triangle/Incenter",
"Definition:Circumcircle of Triangle/Circumcenter",
"Definition:Triangle (Geometry)",
"Definition:Circumcircle of Triangle/Circumradius",
"Definition:Incircle of Triangle/Inradius"
] | [
"Euler Triangle Formula/Lemma 1",
"Euler Triangle Formula/Lemma 2",
"File:Incenter Circumcenter Distance 2.png",
"Definition:Circle/Diameter",
"Thales' Theorem",
"Equal Angles in Equal Circles",
"Equiangular Right Triangles are Similar",
"Difference of Two Squares"
] |
proofwiki-1791 | Open Ball of Point Inside Open Ball/Metric Space | Let $M = \struct {A, d}$ be a metric space.
Let $\map {B_\epsilon} x$ be an open $\epsilon$-ball in $M = \struct {A, d}$.
Let $y \in \map {B_\epsilon} x$.
Then:
:$\exists \delta \in \R: \map {B_\delta} y \subseteq \map {B_\epsilon} x$
That is, for every point in an open $\epsilon$-ball in a metric space, there exists a... | Let $\delta = \epsilon - \map d {x, y}$.
From the definition of open ball, this is strictly positive, since $y \in \map {B_\epsilon} x$.
If $z \in \map {B_\delta} y$, then $\map d {y, z} < \delta$.
So:
:$\map d {x, z} \le \map d {x, y} + \map d {y, z} < \map d {x, y} + \delta = \epsilon$
Thus $z \in \map {B_\epsilon} x... | Let $M = \struct {A, d}$ be a [[Definition:Metric Space|metric space]].
Let $\map {B_\epsilon} x$ be an [[Definition:Open Ball of Metric Space|open $\epsilon$-ball]] in $M = \struct {A, d}$.
Let $y \in \map {B_\epsilon} x$.
Then:
:$\exists \delta \in \R: \map {B_\delta} y \subseteq \map {B_\epsilon} x$
That is, f... | Let $\delta = \epsilon - \map d {x, y}$.
From the definition of [[Definition:Open Ball of Metric Space|open ball]], this is [[Definition:Strictly Positive|strictly positive]], since $y \in \map {B_\epsilon} x$.
If $z \in \map {B_\delta} y$, then $\map d {y, z} < \delta$.
So:
:$\map d {x, z} \le \map d {x, y} + \map ... | Open Ball of Point Inside Open Ball/Metric Space | https://proofwiki.org/wiki/Open_Ball_of_Point_Inside_Open_Ball/Metric_Space | https://proofwiki.org/wiki/Open_Ball_of_Point_Inside_Open_Ball/Metric_Space | [
"Open Balls",
"Open Ball of Point Inside Open Ball",
"Metric Spaces"
] | [
"Definition:Metric Space",
"Definition:Open Ball",
"Definition:Open Ball",
"Definition:Metric Space",
"Definition:Open Ball",
"Definition:Open Ball"
] | [
"Definition:Open Ball",
"Definition:Strictly Positive"
] |
proofwiki-1792 | Open Sets in Metric Space | Let $M = \struct {A, d}$ be a metric space.
Then $\O$ and $A$ are both open in $M$. | We have the results:
:Empty Set is Open in Metric Space
:Metric Space is Open in Itself
{{qed}} | Let $M = \struct {A, d}$ be a [[Definition:Metric Space|metric space]].
Then $\O$ and $A$ are both [[Definition:Open Set (Metric Space)|open]] in $M$. | We have the results:
:[[Empty Set is Open in Metric Space]]
:[[Metric Space is Open in Itself]]
{{qed}} | Open Sets in Metric Space | https://proofwiki.org/wiki/Open_Sets_in_Metric_Space | https://proofwiki.org/wiki/Open_Sets_in_Metric_Space | [
"Open Sets (Metric Spaces)",
"Metric Spaces"
] | [
"Definition:Metric Space",
"Definition:Open Set/Metric Space"
] | [
"Empty Set is Open in Metric Space",
"Metric Space is Open in Itself"
] |
proofwiki-1793 | Equivalence of Definitions of Continuity on Metric Spaces | Let $M_1 = \struct {A_1, d_1}$ and $M_2 = \struct {A_2, d_2}$ be metric spaces.
Let $f: A_1 \to A_2$ be a mapping from $A_1$ to $A_2$.
{{TFAE|def = Continuous on Metric Space|view = continuity|context = Metric Space|contextview = metric spaces}} | === Definition by Points implies Definition by Open Sets ===
Suppose that $f$ is continuous at every point $x \in A_1$.
Let $U \subseteq M_2$ be open in $M_2$.
Let $x \in f^{-1} \sqbrk U$.
Since $U$ is open in $M_2$:
:$\exists \epsilon \in \R_{>0}: \map {B_\epsilon} {\map f x; d_2} \subseteq U$
where $\map {B_\epsilon}... | Let $M_1 = \struct {A_1, d_1}$ and $M_2 = \struct {A_2, d_2}$ be [[Definition:Metric Space|metric spaces]].
Let $f: A_1 \to A_2$ be a [[Definition:Mapping|mapping]] from $A_1$ to $A_2$.
{{TFAE|def = Continuous on Metric Space|view = continuity|context = Metric Space|contextview = metric spaces}} | === Definition by Points implies Definition by Open Sets ===
Suppose that $f$ is [[Definition:Continuous at Point of Metric Space|continuous at every point]] $x \in A_1$.
Let $U \subseteq M_2$ be [[Definition:Open Set (Metric Space)|open]] in $M_2$.
Let $x \in f^{-1} \sqbrk U$.
Since $U$ is [[Definition:Open Set (... | Equivalence of Definitions of Continuity on Metric Spaces | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Continuity_on_Metric_Spaces | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Continuity_on_Metric_Spaces | [
"Continuous Mappings on Metric Spaces"
] | [
"Definition:Metric Space",
"Definition:Mapping"
] | [
"Definition:Continuous Mapping (Metric Space)/Point",
"Definition:Open Set/Metric Space",
"Definition:Open Set/Metric Space",
"Definition:Open Ball",
"Definition:Continuous Mapping (Metric Space)/Point/Definition 3",
"Definition:Open Set/Metric Space",
"Definition:Open Set/Metric Space",
"Definition:O... |
proofwiki-1794 | Union of Open Sets of Metric Space is Open | Let $M = \struct {A, d}$ be a metric space.
The union of a set of open sets of $M$ is open in $M$. | Let $I$ be any indexing set.
Let $U_i$ be open in $M$ for all $i \in I$.
Let $\ds x \in \bigcup_{i \mathop \in I} U_i$.
Then by definition of set union, $x \in U_k$ for some $k \in I$.
Since $U_k$ is open in $M$:
:$\ds \exists \epsilon > 0: \map {B_\epsilon} x \subseteq U_k$
where $\map {B_\epsilon} x$ is the open $\ep... | Let $M = \struct {A, d}$ be a [[Definition:Metric Space|metric space]].
The [[Definition:Set Union|union]] of a [[Definition:Set|set]] of [[Definition:Open Set (Metric Space)|open sets]] of $M$ is [[Definition:Open Set (Metric Space)|open in $M$]]. | Let $I$ be any [[Definition:Indexing Set|indexing set]].
Let $U_i$ be [[Definition:Open Set (Metric Space)|open in $M$]] for all $i \in I$.
Let $\ds x \in \bigcup_{i \mathop \in I} U_i$.
Then by [[Definition:Union of Family|definition of set union]], $x \in U_k$ for some $k \in I$.
Since $U_k$ is [[Definition:Ope... | Union of Open Sets of Metric Space is Open | https://proofwiki.org/wiki/Union_of_Open_Sets_of_Metric_Space_is_Open | https://proofwiki.org/wiki/Union_of_Open_Sets_of_Metric_Space_is_Open | [
"Open Sets (Metric Spaces)",
"Set Union"
] | [
"Definition:Metric Space",
"Definition:Set Union",
"Definition:Set",
"Definition:Open Set/Metric Space",
"Definition:Open Set/Metric Space"
] | [
"Definition:Indexing Set",
"Definition:Open Set/Metric Space",
"Definition:Set Union/Family of Sets",
"Definition:Open Set/Metric Space",
"Definition:Open Ball",
"Set is Subset of Union",
"Subset Relation is Transitive",
"Definition:Open Set/Metric Space"
] |
proofwiki-1795 | Lipschitz Equivalent Metric Spaces are Homeomorphic | Let $M_1 = \struct {A_1, d_1}$ and $M_2 = \struct {A_2, d_2}$ be metric spaces.
Let $M_1$ and $M_2$ be Lipschitz equivalent.
Then $M_1$ and $M_2$ are homeomorphic. | Let $M_1$ and $M_2$ be Lipschitz equivalent.
Then, by definition, $\exists h, k \in \R_{>0}$ such that:
:$\forall x, y \in A_1: h \map {d_1} {x, y} \le \map {d_2} {\map f x, \map f y} \le k \map {d_1} {x, y}$
From the definition of open $\epsilon$-ball:
{{begin-eqn}}
{{eqn | l = y
| o = \in
| r = \map {B_{h... | Let $M_1 = \struct {A_1, d_1}$ and $M_2 = \struct {A_2, d_2}$ be [[Definition:Metric Space|metric spaces]].
Let $M_1$ and $M_2$ be [[Definition:Lipschitz Equivalent Metric Spaces|Lipschitz equivalent]].
Then $M_1$ and $M_2$ are [[Definition:Homeomorphic Metric Spaces|homeomorphic]]. | Let $M_1$ and $M_2$ be [[Definition:Lipschitz Equivalent Metric Spaces|Lipschitz equivalent]].
Then, by definition, $\exists h, k \in \R_{>0}$ such that:
:$\forall x, y \in A_1: h \map {d_1} {x, y} \le \map {d_2} {\map f x, \map f y} \le k \map {d_1} {x, y}$
From the definition of [[Definition:Open Ball of Metric Sp... | Lipschitz Equivalent Metric Spaces are Homeomorphic | https://proofwiki.org/wiki/Lipschitz_Equivalent_Metric_Spaces_are_Homeomorphic | https://proofwiki.org/wiki/Lipschitz_Equivalent_Metric_Spaces_are_Homeomorphic | [
"Lipschitz Equivalence",
"Homeomorphisms (Metric Spaces)"
] | [
"Definition:Metric Space",
"Definition:Lipschitz Equivalence/Metric Spaces",
"Definition:Homeomorphism/Metric Spaces"
] | [
"Definition:Lipschitz Equivalence/Metric Spaces",
"Definition:Open Ball",
"Definition:Open Set/Metric Space",
"Definition:Open Set/Metric Space",
"Definition:Open Set/Metric Space",
"Definition:Open Set/Metric Space",
"Definition:Homeomorphism/Metric Spaces"
] |
proofwiki-1796 | P-Product Metrics on Real Vector Space are Topologically Equivalent | For $n \in \N$, let $\R^n$ be an Euclidean space.
Let $p \in \R_{\ge 1}$.
Let $d_p$ be the $p$-product metric on $\R^n$.
Let $d_\infty$ be the Chebyshev distance on $\R^n$.
Then $d_p$ and $d_\infty$ are topologically equivalent. | Let $r, t \in \R_{\ge 1}$.
{{WLOG}}, assume that $r \le t$.
For all $x, y \in \R^n$, we are going to show that:
:$\map {d_r} {x, y} \ge \map {d_\infty} {x, y} \ge n^{-1} \map {d_r} {x, y}$
Then we can demonstrate Lipschitz equivalence between all of these metrics, from which topological equivalence follows.
Let $d_r$ b... | For $n \in \N$, let $\R^n$ be an [[Definition:Euclidean Space|Euclidean space]].
Let $p \in \R_{\ge 1}$.
Let $d_p$ be the [[Definition:P-Product Metric on Real Vector Space|$p$-product metric]] on $\R^n$.
Let $d_\infty$ be the [[Definition:Chebyshev Distance on Real Vector Space|Chebyshev distance]] on $\R^n$.
The... | Let $r, t \in \R_{\ge 1}$.
{{WLOG}}, assume that $r \le t$.
For all $x, y \in \R^n$, we are going to show that:
:$\map {d_r} {x, y} \ge \map {d_\infty} {x, y} \ge n^{-1} \map {d_r} {x, y}$
Then we can demonstrate [[Definition:Lipschitz Equivalent Metrics|Lipschitz equivalence]] between all of these metrics, from wh... | P-Product Metrics on Real Vector Space are Topologically Equivalent | https://proofwiki.org/wiki/P-Product_Metrics_on_Real_Vector_Space_are_Topologically_Equivalent | https://proofwiki.org/wiki/P-Product_Metrics_on_Real_Vector_Space_are_Topologically_Equivalent | [
"P-Product Metrics",
"P-Product Metrics on Real Vector Space are Topologically Equivalent"
] | [
"Definition:Euclidean Space",
"Definition:P-Product Metric/Real Vector Space",
"Definition:Chebyshev Distance/Real Vector Space",
"Definition:Topologically Equivalent Metrics"
] | [
"Definition:Lipschitz Equivalence/Metrics",
"Lipschitz Equivalent Metrics are Topologically Equivalent",
"Definition:Metric Space/Metric",
"Definition:Lipschitz Equivalence/Metrics"
] |
proofwiki-1797 | Derivative of Function of Constant Multiple | Let $f$ be a real function which is differentiable on $\R$.
Let $c \in \R$ be a constant.
Then:
:$\map {D_x} {\map f {c x} } = c \map {D_{c x} } {\map f {c x} }$ | First it is shown that $\map {D_x} {c x} = c$:
{{begin-eqn}}
{{eqn | l = \map {D_x} {c x}
| r = c \map {D_x} x + x \map {D_x} c
| c = Product Rule for Derivatives
}}
{{eqn | r = c + x \map {D_x} c
| c = Derivative of Identity Function
}}
{{eqn | r = c + 0
| c = Derivative of Constant
}}
{{eqn | ... | Let $f$ be a [[Definition:Real Function|real function]] which is [[Definition:Differentiable on Interval|differentiable]] on $\R$.
Let $c \in \R$ be a constant.
Then:
:$\map {D_x} {\map f {c x} } = c \map {D_{c x} } {\map f {c x} }$ | First it is shown that $\map {D_x} {c x} = c$:
{{begin-eqn}}
{{eqn | l = \map {D_x} {c x}
| r = c \map {D_x} x + x \map {D_x} c
| c = [[Product Rule for Derivatives]]
}}
{{eqn | r = c + x \map {D_x} c
| c = [[Derivative of Identity Function]]
}}
{{eqn | r = c + 0
| c = [[Derivative of Constant]... | Derivative of Function of Constant Multiple | https://proofwiki.org/wiki/Derivative_of_Function_of_Constant_Multiple | https://proofwiki.org/wiki/Derivative_of_Function_of_Constant_Multiple | [
"Derivative of Function of Constant Multiple",
"Derivative of Constant Multiple",
"Derivatives"
] | [
"Definition:Real Function",
"Definition:Differentiable Mapping/Real Function/Interval"
] | [
"Product Rule for Derivatives",
"Derivative of Identity Function",
"Derivative of Constant",
"Derivative of Composite Function"
] |
proofwiki-1798 | Cotangent Function is Periodic on Reals | The real cotangent function is periodic with period $\pi$. | {{begin-eqn}}
{{eqn | l = \map \cot {x + \pi}
| r = \frac {\map \cos {x + \pi} } {\map \sin {x + \pi} }
| c = {{Defof|Cotangent}}
}}
{{eqn | r = \frac {-\cos x} {-\sin x}
| c = Cosine of Angle plus Straight Angle, Sine of Angle plus Straight Angle
}}
{{eqn | r = \cot x
| c =
}}
{{end-eqn}}
Also... | The [[Definition:Real Cotangent Function|real cotangent function]] is [[Definition:Periodic Real Function|periodic]] with [[Definition:Period of Periodic Real Function|period $\pi$]]. | {{begin-eqn}}
{{eqn | l = \map \cot {x + \pi}
| r = \frac {\map \cos {x + \pi} } {\map \sin {x + \pi} }
| c = {{Defof|Cotangent}}
}}
{{eqn | r = \frac {-\cos x} {-\sin x}
| c = [[Cosine of Angle plus Straight Angle]], [[Sine of Angle plus Straight Angle]]
}}
{{eqn | r = \cot x
| c =
}}
{{end-eq... | Cotangent Function is Periodic on Reals | https://proofwiki.org/wiki/Cotangent_Function_is_Periodic_on_Reals | https://proofwiki.org/wiki/Cotangent_Function_is_Periodic_on_Reals | [
"Cotangent Function"
] | [
"Definition:Cotangent/Real Function",
"Definition:Periodic Function/Real",
"Definition:Periodic Real Function/Period"
] | [
"Cosine of Angle plus Straight Angle",
"Sine of Angle plus Straight Angle",
"Derivative of Cotangent Function",
"Shape of Sine Function",
"Definition:Strictly Positive Real Function",
"Definition:Real Interval/Open",
"Derivative of Monotone Function",
"Definition:Strictly Decreasing",
"Definition:Pe... |
proofwiki-1799 | Shape of Cotangent Function | The nature of the cotangent function on the set of real numbers $\R$ is as follows:
:$\cot x$ is continuous and strictly decreasing on the interval $\openint 0 \pi$
:$\cot x \to +\infty$ as $x \to 0^+$
:$\cot x \to -\infty$ as $x \to \pi^-$
:$\cot x$ is not defined on $\forall n \in \Z: x = n \pi$, at which points it i... | $\cot x$ is continuous and strictly decreasing on $\openint 0 \pi$:
Continuity follows from the Quotient Rule for Continuous Real Functions:
:$(1): \quad$ Both $\sin x$ and $\cos x$ are continuous on $\openint 0 \pi$ from Real Sine Function is Continuous and Cosine Function is Continuous
:$(2): \quad \sin x > 0$ on thi... | The nature of the [[Definition:Cotangent|cotangent]] function on the set of [[Definition:Real Number|real numbers]] $\R$ is as follows:
:$\cot x$ is [[Definition:Continuous on Interval|continuous]] and [[Definition:Strictly Decreasing Real Function|strictly decreasing]] on the [[Definition:Open Real Interval|interval]... | $\cot x$ is [[Definition:Continuous on Interval|continuous]] and [[Definition:Strictly Decreasing Real Function|strictly decreasing]] on $\openint 0 \pi$:
Continuity follows from the [[Quotient Rule for Continuous Real Functions]]:
:$(1): \quad$ Both $\sin x$ and $\cos x$ are [[Definition:Continuous on Interval|conti... | Shape of Cotangent Function | https://proofwiki.org/wiki/Shape_of_Cotangent_Function | https://proofwiki.org/wiki/Shape_of_Cotangent_Function | [
"Cotangent Function"
] | [
"Definition:Cotangent",
"Definition:Real Number",
"Definition:Continuous Real Function/Interval",
"Definition:Strictly Decreasing/Real Function",
"Definition:Real Interval/Open",
"Definition:Discontinuous"
] | [
"Definition:Continuous Real Function/Interval",
"Definition:Strictly Decreasing/Real Function",
"Combination Theorem for Continuous Functions/Real/Quotient Rule",
"Definition:Continuous Real Function/Interval",
"Real Sine Function is Continuous",
"Cosine Function is Continuous",
"Definition:Strictly Dec... |
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